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THEORY OF 
PLATES AND SHELLS 



S. TIMOSHENKO 

Professor Emeritus of Engineering Mechanics 
Stanford University 



S. WOINOWSKY-KRIEGER 

Professor of Engineering Mechanics CATAUOGOH* 
Laval University 



Second Edition 



McGRAW-HILI 

CLASSIC 
TEXTB®K 
REISSUE 



LIBRARY 

24 JUL 1989 

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Timoshenko and Gere : Theory of Elastic Stability 

Timoshenko and Goodier: Theory of Elasticity 

Timoshenko and Woinowsky-Krieger: Theory of Plates and Shells 

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Papoulis: The Fourier Integral and its Applications 
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Timoshenko: Theory of Plates and Shells 
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THEORY OF PLATES AND SHELLS 

Copyright © 1959 by the McGraw-Hill Book Company, Inc. Reissued 1987 by the McGraw- 
Hill Book Company, Inc. All rights reserved. Printed in the United States of America. 
Except as permitted under the United States Copyright Act of 1976, no part of this pub- 
lication may be reproduced or distributed in any form or by any means, or stored in a data 
base or retrieval system, without the prior written permission of the publisher. 

ISBN 0-07-0^77^-3 

30 31 32 33 34 35 VBA VBA 8 9 2 10 9 8 



PREFACE 



Since the publication of the first edition of this book, the application of 
the theory of plates and shells in practice has widened considerably, and 
some new methods have been introduced into the theory. To take these 
facts into consideration, we have had to make many changes and addi- 
tions. The principal additions are (1) an article on deflection of plates due 
to transverse shear, (2) an article on stress concentrations around a cir- 
cular hole in a bent plate, (3) a chapter on bending of plates resting on 
an elastic foundation, (4) a chapter on bending of anisotropic plates, and 
(5) a chapter reviewing certain special and approximate methods used in 
plate analysis. We have also expanded the chapter on large deflections 
of plates, adding several new cases of plates of variable thickness and 
some numerical tables facilitating plate analysis. 

In the part of the book dealing with the theory of shells, we limited 
ourselves to the addition of the stress-function method in the membrane 
theory of shells and some minor additions in the flexural theory of shells. 

The theory of shells has been developing rapidly in recent years, and 
several new books have appeared in this field. Since it was not feasible 
for us to discuss these new developments in detail, we have merely re- 
ferred to the new bibliography, in which persons specially interested in 
this field will find the necessary information. 

S. Timoshenko 

S. Woinowsky-Krieger 



CONTENTS 

Preface v 

Notation xiu 

Introduction 1 

Chapter 1. Bending of Long Rectangular Plates to a Cylindrical Surface . 4 

1. Differential Equation for Cylindrical Bending of Plates 4 

2. Cylindrical Bending of Uniformly Loaded Rectangular Plates with Simply 
Supported Edges 6 

3. Cylindrical Bending of Uniformly Loaded Rectangular Plates with Built-in 
Edges 13 

4. Cylindrical Bending of Uniformly Loaded Rectangular Plates with Elasti- 
cally Built-in Edges 17 

5. The Effect on Stresses and Deflections of Small Displacements of Longi- 
tudinal Edges in the Plane of the Plate 20 

6. An Approximate Method of Calculating the Parameter u 24 

7. Long Uniformly Loaded Rectangular Plates Having a Small Initial Cylin- 
drical Curvature 27 

8. Cylindrical Bending of a Plate on an Elastic Foundation 30 

Chapter 2. Pure Bending of Plates 33 

9. Slope and Curvature of Slightly Bent Plates 33 

10. Relations between Bending Moments and Curvature in Pure Bending of 
Plates 37 

11. Particular Cases of Pure Bending 42 

12. Strain Energy in Pure Bending of Plates 46 

13. Limitations on the Application of the Derived Formulas 47 

14. Thermal Stresses in Plates with Clamped Edges 49 

Chapter 3. Symmetrical Bending of Circular Plates 51 

15. Differential Equation for Symmetrical Bending of Laterally Loaded Cir- 
cular Plates 51 

16. Uniformly Loaded Circular Plates 54 

17. Circular Plate with a Circular Hole at the Center 58 

18. Circular Plate Concentrically Loaded • 63 

19. Circular Plate Loaded at the Center 67 

20. Corrections to the Elementary Theory of Symmetrical Bending of Cir- 
cular Plates 72 

Chapter 4. Small Deflections of Laterally Loaded Plates 79 

21. The Differential Equation of the Deflection Surface . 7% 

vii 



Vlll CONTENTS 

22. Boundary Conditions 83 

23. Alternate Method of Derivation of the Boundary Conditions .... 88 

24. Reduction of the Problem of Bending of a Plate to That of Deflection of a 
Membrane 92 

25. Effect of Elastic Constants on the Magnitude of Bending Moments . . 97 

26. Exact Theory of Plates 98 

Chapter 6. Simply Supported Rectangular Plates 105 

27. Simply Supported Rectangular Plates under Sinusoidal Load .... 105 

28. Navier Solution for Simply Supported Rectangular Plates 108 

29. Further Applications of the Navier Solution Ill 

30. Alternate Solution for Simply Supported and Uniformly Loaded Rectangu- 
lar Plates H3 

31. Simply Supported Rectangular Plates under Hydrostatic Pressure . 124 

32. Simply Supported Rectangular Plate under a Load in the Form of a Tri- 
angular Prism 130 

33. Partially Loaded Simply Supported Rectangular Plate 135 

34. Concentrated Load on a Simply Supported Rectangular Plate .... 141 

35. Bending Moments in a Simply Supported Rectangular Plate with a Con- 
centrated Load 143 

36. Rectangular Plates of Infinite Length with Simply Supported Edges . . 149 

37. Bending Moments in Simply Supported Rectangular Plates under a Load 
Uniformly Distributed over the Area of a Rectangle 158 

38. Thermal Stresses in Simply Supported Rectangular Plates 162 

39. The Effect of Transverse Shear Deformation on the Bending of Thin Plates 165 

40. Rectangular Plates of Variable Thickness 173 

Chapter 6. Rectangular Plates with Various Edge Conditions 180 

41. Bending of Rectangular Plates by Moments Distributed along the Edges . 180 

42. Rectangular Plates with Two Opposite Edges Simply Supported and the 
Other Two Edges Clamped 185 

43. Rectangular Plates with Three Edges Simply Supported and One Edge 
Built In 192 

44. Rectangular Plates with All Edges Built In 197 

45. Rectangular Plates with One Edge or Two Adjacent Edges Simply Sup- 
ported and the Other Edges Built In 205 

46. Rectangular Plates with Two Opposite Edges Simply Supported, the Third 
Edge Free, and the Fourth Edge Built In or Simply Supported . . . 208 

47. Rectangular Plates with Three Edges Built In and the Fourth Edge Free. 211 

48. Rectangular Plates with Two Opposite Edges Simply Supported and the 
Other Two Edges Free or Supported Elastically 214 

49. Rectangular Plates Having Four Edges Supported Elastically or Resting 

on Corner Points with All Edges Free 218 

50. Semi-infinite Rectangular Plates under Uniform Pressure 221 

51. Semi-infinite Rectangular Plates under Concentrated Loads .... 225 

Chapter 7. Continuous Rectangular Plates 229 

52. Simply Supported Continuous Plates 229 

53. Approximate Design of Continuous Plates with Equal Spans .... 236 

54. Bending of Plates Supported by Rows of Equidistant Columns (Flat Slabs) 245 

55. Flat Slab Having Nine Panels and Slab with Two Edges Free .... 253 

56. Effect of a Rigid Connection with Column or- Moments of the Flat Slab . 257 



CONTENTS 



ix 



Chapter 8. Plates on Elastic Foundation 259 

57. Bending Symmetrical with Respect to a Center 259 

58. Application of Bessel Functions to the Problem of the Circular Plate . . 265 

59. Rectangular and Continuous Plates on Elastic Foundation .... 269 

60. Plate Carrying Rows of Equidistant Columns ^ 276 

61. Bending of Plates Resting on a Semi-infinite Elastic Solid 278 

Chapter 9. Plates of Various Shapes 282 

62. Equations of Bending of Plates in Polar Coordinates 282 

63. Circular Plates under a Linearly Varying Load 285 

64. Circular Plates under a Concentrated Load 290 

65. Circular Plates Supported at Several Points along the Boundary ... 293 

66. Plates in the Form of a Sector 295 

67. Circular Plates of Nonuniform Thickness 298 

68. Annular Plates with Linearly Varying Thickness 303 

69. Circular Plates with Linearly Varying Thickness 305 

70. Nonlinear Problems in Bending of Circular Plates < 308 

71. Elliptical Plates 310 

72. Triangular Plates 313 

73. Skewed Plates 318 

74. Stress Distribution around Holes 319 

Chapter 10. Special and Approximate Methods in Theory of Plates ... 325 

75. Singularities in Bending of Plates 325 

76. The Use of Influence Surfaces in the Design of Plates 328 

77. Influence Functions and Characteristic Functions 334 

78. The Use of Infinite Integrals and Transforms 336 

79. Complex Variable Method 340 

80. Application of the Strain Energy Method in Calculating Deflections . . 342 

81. Alternative Procedure in Applying the Strain Energy Method .... 347 

82. Various Approximate Methods 348 

83. Application of Finite Differences Equations to the Bending of Simply Sup- 
ported Plates 351 

84. Experimental Methods 362 

Chapter 11. Bending of Anisotropic Plates 364 

85. Differential Equation of the Bent Plate 364 

86. Determination of Rigidities in Various Specific Cases 366 

87. Application of the Theory to the Calculation of Gridworks .... 369 

88. Bending of Rectangular Plates 371 

89. Bending of Circular and Elliptic Plates 376 

Chapter 12. Bending of Plates under the Combined Action of Lateral Loads 

and Forces in the Middle Plane of the Plate 378 

90. Differential Equation of the Deflection Surface 378 

91. Rectangular Plate with Simply Supported Edges under the Combined 
Action of Uniform Lateral Load and Uniform Tension 380 

92. Application of the Energy Method . 382 

93. Simply Supported Rectangular Plates under the Combined Action of 
Lateral Loads and of Forces in the Middle Plane of the Plate . . .387 

94. Circular Plates under Combined Action of Lateral Load and Tension or 

OQ1 

Compression OXJ1 

95. Bending of Plates with a Small Initial Curvature 393 



X CONTENTS 

Chapter 13. Large Deflections of Plates 396 

96. Bending of Circular Plates by Moments Uniformly Distributed along the 
Edge 396 

97. Approximate Formulas for Uniformly Loaded Circular Plates with Large 
Deflections 400 

98. Exact Solution for a Uniformly Loaded Circular Plate with a Clamped 
Edge 404 

99. A Simply Supported Circular Plate under Uniform Load 408 

100. Circular Plates Loaded at the Center 412 

101. General Equations for Large Deflections of Plates 415 

102. Large Deflections of Uniformly Loaded Rectangular Plates . . . .421 

103. Large Deflections of Rectangular Plates with Simply Supported Edges . 425 

Chapter 14. Deformation of Shells without Bending 429 

104. Definitions and Notation 429 

105. Shells in the Form of a Surface of Revolution and Loaded Symmetrically 
with Respect to Their Axis 433 

106. Particular Cases of Shells in the Form of Surfaces of Revolution . . . 436 

107. Shells of Constant Strength 442 

108. Displacements in Symmetrically Loaded Shells Having the Form of a 
Surface of Revolution 445 

109. Shells in the Form of a Surface of Revolution under Unsymmetrical 
Loading 447 

110. Stresses Produced by Wind Pressure 449 

111. Spherical Shell Supported at Isolated Points 453 

112. Membrane Theory of Cylindrical Shells 457 

113. The Use of a Stress Function in Calculating Membrane Forces of Shells . 461 

Chapter 16. General Theory of Cylindrical Shells 466 

114. A Circular Cylindrical Shell Loaded Symmetrically with Respect to Its Axis 466 

115. Particular Cases of Symmetrical Deformation of Circular Cylindrical Shells 471 

116. Pressure Vessels 481 

117. Cylindrical Tanks with Uniform Wall Thickness 485 

118. Cylindrical Tanks with Nonuniform Wall Thickness 488 

119. Thermal Stresses in Cylindrical Shells 497 

120. Inextensional Deformation of a Circular Cylindrical Shell 501 

121. General Case of Deformation of a Cylindrical Shell 507 

122. Cylindrical Shells with Supported Edges 514 

123. Deflection of a Portion of a Cylindrical Shell 516 

124. An Approximate Investigation of the Bending of Cylindrical Shells . . 519 

125. The Use of a Strain and Stress Function 522 

126. Stress Analysis of Cylindrical Roof Shells 524 

Chapter 16. Shells Having the Form of a Surface of Revolution and Loaded 

Symmetrically with Respect to Their Axis 533 

127. Equations of Equilibrium 533 

128. Reduction of the Equations of Equilibrium to Two Differential Equations 

of the Second Order 537 

129. Spherical Shell of Constant Thickness 540 



CONTENTS XI 

130. Approximate Methods of Analyzing Stresses in Spherical Shells . . . 547 

131. Spherical Shells with an Edge Ring 555 

132. Symmetrical Bending of Shallow Spherical Shells ^ 558 

133. Conical Shells 562 

134. General Case of Shells Having the Form of a Surface of Revolution . . 566 

Name Index 569 

Subject Index 575 



NOTATION 



x, y, z Rectangular coordinates 

r, 6 Polar coordinates 
r x , r y Radii of curvature of the middle surface of a plate in xz and yz planes, 
respectively 
h Thickness of a plate or a shell 
q Intensity of a continuously distributed load 
p Pressure 
P Single load 

Weight per unit volume 

Normal components of stress parallel to x, y, and z axes 
Normal component of stress parallel to n direction 
Radial stress in polar coordinates 
Tangential stress in polar coordinates 
Shearing stress 

Shearing stress components in rectangular coordinates 
u, v, w Components of displacements 
e Unit elongation 

Unit elongations in x, y, and z directions 
Radial unit elongation in polar coordinates 
e h ee Tangential unit elongation in polar coordinates 
€<p , ee Unit elongations of a shell in meridional direction and in the direction 
of parallel circle, respectively 
7xy, y*z, lyz Shearing strain components in rectangular coordinates 
y r e Shearing strain in polar coordinates 
E Modulus of elasticity in tension and compression 
G Modulus of elasticity in shear 
v Poisson's ratio 
V Strain energy 

D Flexural rigidity of a plate or shell 
M x , My Bending moments per unit length of sections of a plate perpendicular 
to x and y axes, respectively 
M xy Twisting moment per unit length of section of a plate perpendicular 
to x axis 
M r „ Mnt Bending and twisting moments per unit length of a section of a plate 
perpendicular to n direction 
Qx, Q y Shearing forces parallel to z axis per unit length of sections of a plate 
perpendicular to x and y axes, respectively 
Q tl Shearing force parallel to z axis per unit length of section of a plate 
perpendicular to n direction 
N x , N y Normal forces per unit length of sections of a plate perpendicular to 
x and y directions, respectively 



T 
» 0"z 



Txy, T X z, T yz 



e r 



XIV NOTATION 

N xy Shearing force in direction of y axis per unit length of section of a plate 
perpendicular to x axis 
M r , Mt, Mh Radial, tangential, and twisting moments when using polar coordinates 
Q rj Q t Radial and tangential shearing forces 

N r , Nt Normal forces per unit length in radial and tangential directions 
n, r 2 Radii of curvature of a shell in the form of a surface of revolution in 
meridional plane and in the normal plane perpendicular to meridian, 
respectively 
X<p f xe Changes of curvature of a shell in meridional plane and in the plane 
perpendicular to meridian, respectively 
xe<p Twist of a shell 
X, Y, Z Components of the intensity of the external load on a shell, parallel to 
x, y, and z axes, respectively 
N<p y Ne, N<pd Membrane forces per unit length of principal normal sections of a shell 
Me, M<p Bending moments in a shell per unit length of meridional section and a 
section perpendicular to meridian, respectively 
Xx, x<p Changes of curvature of a cylindrical shell in axial plane and in a plane 
perpendicular to the axis, respectively 
N*, N x , N x<p Membrane forces per unit length of axial section and a section perpen- 
dicular to the axis of a cylindrical shell 
Mtp, M x Bending moments per unit length of axial section and a section perpen- 
dicular to the axis of a cylindrical shell, respectively 
M Xip Twisting moment per unit length of an axial section of a cylindrical 
shell 
Qip, Qx Shearing forces parallel to z axis per unit length of an axial section and 
a section perpendicular to the axis of a cylindrical shell, respectively 
ibg Natural logarithm 
log 10, Log Common logarithm 



INTRODUCTION 



The bending properties of a plate depend greatly on its thickness as 
compared with its other dimensions. In the following discussion, we 
shall distinguish between three kinds of plates: (1) thin plates with small 
deflections, (2) thin plates with large deflections, (3) thick plates. 

Thin Plates with Small Deflection. If deflections w of a plate are small 
in comparison with its thickness h } el very satisfactory approximate theory 
of bending of the plate by lateral loads can be developed by making the 
following assumptions: 

1. There is no deformation in the middle plane of the plate. This 
plane remains neutral during bending. 

2. Points of the plate lying initially on a normal-to-the-middle plane 
of the plate remain on the normal-to-the-middle surface of the plate after 
bending. 

3. The normal stresses in the direction transverse to the plate can be 
disregarded. 

Using these assumptions, all stress components can be expressed by 
deflection w of the plate, which is a function of the two coordinates in 
the plane of the plate. This function has to satisfy a linear partial 
differential equation, which, together with the boundary conditions, com- 
pletely defines w. Thus the solution of this equation gives all necessary 
information for calculating stresses at any point of the plate. 

The second assumption is equivalent to the disregard of the effect of 
shear forces on the deflection of plates. This assumption is usually satis- 
factory, but in some cases (for example, in the case of holes in a plate) 
the effect of shear becomes important and some corrections in the theory 
of thin plates should be introduced (see Art. 39). 

If, in addition to lateral loads, there are external forces acting in the 
middle plane of the plate, the first assumption does not hold any more, 
and it is necessary to take into consideration the effect on bending of the 
plate of the stresses acting in the middle plane of the plate. This can be 
done by introducing some additional terms into the above-mentioned 
differential equation of plates (see Art. 90). 

1 



2 THEORY OF PLATES AND SHELLS 

Thin Plates with Large Deflection. The first assumption is completely 
satisfied only if a plate is bent into a developable surface. In other cases 
bending of a plate is accompanied by strain in the middle plane, but 
calculations show that the corresponding stresses in the middle plane are 
negligible if the deflections of the plate are small in comparison with its 
thickness. If the deflections are not small, these supplementary stresses 
must be taken into consideration in deriving the differential equation of 
plates. In this way we obtain nonlinear equations and the solution of the 
problem becomes much more complicated (see Art. 96). In the case of 
large deflections we have also to distinguish between immovable edges 
and edges free to move in the plane of the plate, which may have a con- 
siderable bearing upon the magnitude of deflections and stresses of the 
plate (see Arts. 99, 100). Owing to the curvature of the deformed middle 
plane of the plate, the supplementary tensile stresses, which predominate, 
act in opposition to the given lateral load; thus, the given load is now 
transmitted partly by the flexural rigidity and partly by a membrane 
action of the plate. Consequently, very thin plates with negligible 
resistance to bending behave as membranes, except perhaps for a narrow 
edge zone where bending may occur because of the boundary conditions 
imposed on the plate. 

The case of a plate bent into a developable, in particular into a cylindri- 
cal, surface should be considered as an exception. The deflections of 
such a plate may be of the order of its thickness without necessarily pro- 
ducing membrane stresses and without affecting the linear character of 
the theory of bending. Membrane stresses would, however, arise in such 
a plate if its edges are immovable in its plane and the deflections are 
sufficiently large (see Art. 2). Therefore, in " plates with small deflec- 
tion" membrane forces caused by edges immovable in the plane of the 
plate can be practically disregarded. 

Thick Plates. The approximate theories of thin plates, discussed 
above, become unreliable in the case of plates of considerable thickness, 
especially in the case of highly concentrated loads. In such a case the 
thick-plate theory should be applied. This theory considers the prob- 
lem of plates as a three-dimensional problem of elasticity. The stress 
analysis becomes, consequently, more involved and, up to now, the prob- 
lem is completely solved only for a few particular cases. Using this 
analysis, the necessary corrections to the thin-plate theory at the points of 
application of concentrated loads can be introduced. 

The main suppositions of the theory of thin plates also form the basis 
for the usual theory of thin shells. There exists, however, a substantial 
difference in the behavior of plates and shells under the action of external 
loading. The static equilibrium of a plate element under a lateral load 
is only possible by action of bending and twisting moments, usually 



INTRODUCTION 6 

accompanied by shearing forces, while a shell, in general, is able to trans- 
mit the surface load by "membrane" stresses which act parallel to the 
tangential plane at a given point of the middle surface and are distributed 
uniformly over the thickness of the shell. This property of shells makes 
them, as a rule, a much more rigid and a more economical structure than 
a plate would be under the same conditions. 

In principle, the membrane forces are independent of bending and are 
wholly defined by the conditions of static equilibrium. The methods of 
determination of these forces represent the so-called "membrane theory 
of shells." However, the reactive forces and deformation obtained by 
the use of the membrane theory at the shell's boundary usually become 
incompatible with the actual boundary conditions. To remove this dis- 
crepancy the bending of the shell in the edge zone has to be considered, 
which may affect slightly the magnitude of initially calculated membrane 
forces. This bending, however, usually has a very localized 1 character 
and may be calculated on the basis of the same assumptions which were 
used in the case of small deflections of thin plates. But there are prob- 
lems, especially those concerning the elastic stability of shells, in which 
the assumption of small deflections should be discontinued and the "large- 
deflection theory" should be used. 

If the thickness of a shell is comparable to the radii of curvature, or 
if we consider stresses near the concentrated forces, a more rigorous 
theory, similar to the thick-plate theory, should be applied. 

1 There are some kinds of shells, especially those with a negative Gaussian curva- 
ture, which provide us with a lot of exceptions. In the case of developable surfaces 
such as cylinders or cones, large deflection without strain of the middle surface is 
possible, and, in some cases, membrane stresses can be neglected and consideration 
of the bending stresses alone may be sufficient. 



CHAPTER 1 

BENDING OF LONG RECTANGULAR PLATES TO A 
CYLINDRICAL SURFACE 



1. Differential Equation for Cylindrical Bending of Plates. We shall 
begin the theory of bending of plates with the simple problem of the 
bending of a long rectangular plate that is subjected to a transverse load 
that does not vary along the length of the plate. The deflected surface 
of a portion of such a plate at a considerable distance from the ends 1 
can be assumed cylindrical, with the axis of the cylinder parallel to the 
length of the plate. We can therefore restrict ourselves to the investi- 
gation of the bending of an elemental strip cut from the plate by two 
planes perpendicular to the length of the plate and a unit distance (say 
1 in.) apart. The deflection of this strip is given by a differential equa- 
tion which is similar to the deflection 
equation of a bent beam. 

To obtain the equation for the de- 
flection, we consider a plate of uni- 
form thickness, equal to h, and take 
the xy plane as the middle plane of 
the plate before loading, i.e., as the 
plane midway between the faces of 
the plate. Let the y axis coincide with one of the longitudinal edges 
of the plate and let the positive direction of the z axis be downward, 
as shown in Fig. 1. Then if the width of the plate is denoted by Z, the 
elemental strip may be considered as a bar of rectangular cross section 
which has a length of I and a depth of h. In calculating the bending 
stresses in such a bar we assume, as in the ordinary theory of beams, 
that cross sections of the bar remain plane during bending, so that they 
undergo only a rotation with respect to their neutral axes. If no normal 
forces are applied to the end sections of the bar, the neutral surface of 
the bar coincides with the middle surface of the plate, and the unit 
elongation of a fiber parallel to the x axis is proportional to its distance z 

1 The relation between the length and the width of a plate in order that the maxi- 
mum stress may approximate that in an infinitely long plate is discussed later, see 
pp. 118 and 125. 

4 




BENDING TO A CYLINDRICAL SURFACE 



from the middle surface. The curvature of the deflection curve can be 
taken equal to —d 2 w/dx 2 , where w, the deflection of the bar in the z 
direction, is assumed to be small compared with the length of the bar I. 
The unit elongation e x of a fiber at a distance z from the middle surface 
(Fig. 2) is then -z d 2 w/dx 2 . 

Making use of Hooke's law, the unit elonga- 
tions e x and e y in terms of the normal stresses 
a x and <r y acting on the element shown shaded 
in Fig. 2a are 




E 



E 



E 

Vdx 

~E 



try (a) 

,f ttHtf, 



(1) 



- 4? = 



-or x 



TTTTTT 
(b) 

Fig. 2 



where E is the modulus of elasticity of the 

material and v is Poisson's ratio. The lateral 

strain in the y direction must be zero in order to maintain continuity 

in the plate during bending, from which it follows by the second of the 

equations (1) that a y = va x . Substituting this value in the first of the 

equations (I), we obtain 



and 



_ (1 - V 2 )<T X 

E 
Ee x = _ Ez d 2 w 

1 _ v 2 JZ2 



1 - 



(2) 



If the plate is submitted to the action of tensile or compressive forces 
acting in the x direction and uniformly distributed along the longitudinal 
sides of the plate, the corresponding direct stress must be added to the 
stress (2) due to bending. 

Having the expression for bending stress <r x , we obtain by integration 
the bending moment in the elemental strip : 



M 



fh/2 r 

J-h/2 J- 



Introducing the notation 



h/2 £ Z 2 
-h/2 1 — V 

Eh' 



d 2 w 



! dx 2 



dz 



Eh* d 2 w 



12(1 - v 2 ) dx 2 



= D 



12(1 - v 2 ) " (3) 

we represent the equation for the deflection curve of the elemental strip 
in the following form : 

(4) 



c £ = -« 



in which the quantity D, taking the place of the quantity EI in the case 



6 THEOKY OF PLATES AND SHELLS 

of beams, is called the flexural rigidity of the plate. It is seen that the 
calculation of deflections of the plate reduces to the integration of Eq. (4), 
which has the same form as the differential equation for deflection of 
beams. If there is only a lateral load acting on the plate and the edges 
are free to approach each other as deflection occurs, the expression for 
the bending moment M can be readily derived, and the deflection curve 
is then obtained by integrating Eq. (4). In practice the problem is more 
complicated, since the plate is usually attached to the boundary and its 
edges are not free to move. Such a method of support sets up tensile 
reactions along the edges as soon as deflection takes place. These reac- 
tions depend on the magnitude of the deflection and affect the magnitude 
of the bending moment M entering in Eq. (4). The problem reduces to 
the investigation of bending of an elemental strip submitted to the action 
of a lateral load and also an axial force which depends on the deflection 
of the strip. 1 In the following we consider this problem for the particular 
case of uniform load acting on a plate and for various conditions along 
the edges. 

2. Cylindrical Bending of Uniformly Loaded Rectangular Plates with 
Simply Supported Edges. Let us consider a uniformly loaded long rec- 
tangular plate with longitudinal edges which are free to rotate but can- 
not move toward each other during bending. An elemental strip cut out 



H^iQjiiik^ 



<— x-— > 



Fig. 3 



from this plate, as shown in Fig. 1, is in the condition of a uniformly 
loaded bar submitted to the action of an axial force S (Fig. 3). The 
magnitude of S is such as to prevent the ends of the bar from moving 
along the x axis. Denoting by q the intensity of the uniform load, the 
bending moment at any cross section of the strip is 






Sw 



1 In such a form the problem was first discussed by I. G. Boobnov; see the English 
translation of his work in Trans. Inst. Naval Architects, vol. 44, p. 15, 1902, and his 
"Theory of Structure of Ships," vol. 2, p. 545, St. Petersburg, 1914. See also the 
paper by Stewart Way presented at the National Meeting of Applied Mechanics, 
ASME, New Haven, Conn., June, 1932; from this paper are taken the curves used in 
Arts. 2 and 3. 



BENDING TO A CYLINDRICAL SURFACE 

Substituting in Eq. (4), we obtain 



Introducing the notation 



the general solution of Eq. (a) can be written in the following <3rm: 



d 2 w 

dx^ ' 


Sw qlx qx 2 
" ~D ~ 2D + 2D 


on 


S I* 2 
D± = U 



„ . , 2ux , „ , 2ux . ql 3 x 
w = C i smh — -j h C 2 cosh — = h ^ 2 ^ 



ql 2 x 2 

Su 2 D " 16w"I> 



<?Z 



(a) 



(5) 



(&) 



The constants of integration C\ and C 2 will be determined from the 
conditions at the ends. Since the deflections of the strip at th(; ends are 
zero, we have 

w = for x = and x = Z (c) 

Substituting for w its expression (6), we obtain from these two c< editions 

ql A 1 — cosh 2u n _ ql* 



Ci = 



c 2 = 



16u A D sinh 2u "* 16m 4 Z> 

and the expression (6) for the deflection w becomes 



w = 



ff z- 



4 D\ 



16w 
Substituting 



— cosh 2u . , 2w# . , 2ux 

. . ^ smh — ; h cosh — 7— 

smh 2u I I 







1 + 



g/ 3 x* g/V 2 



8w 2 £» 8w 2 D 



cosh 2u = cosh 2 w + sinh 2 u sinh 2w = 2 sinh w cosh w 
cosh 2 u = 1 + sinh 2 u 



we can represent this expression in a simpler form: 
ql* 



w = 



or 



. , . , 2ux . . , 2ux 

smh w smh — f- cosh u cosh —j- 



16u*D 



cosh u 



>+&«-> 



w = 



g/ 4 



16u 4 Z> 



cosh w 

cosh u 



Ml, 



+ 8u 2 D ^ 



x) 



(6) 



Thus, deflections of the elemental strip depend upon the quantity u, 
which, as we see from Eq. (5), is a function of the axial force S. This 
force can be determined from the condition that the ends of the strip 
(Fig. 3) do not move along the x axis. Hence the extension of the strip 
produced by the forces S is equal to the difference between the length of 
the arc along the deflection curve and the chord length I. This difference 



8 THEORY OF PLATES AND SHELLS 

for small deflections can be represented by the formula 1 



-U'(rl) 



dx (7) 



In calculating the extension of the strip produced by the forces S. we 
assume that the lateral strain of the strio in the y direction is prevented 
and use Eq. (2). Then 



Substituting expression (6) for w and performing the integration, wc 
obtain the following equation for calculating S: 

S(l - v 2 )l _ q 2 



hE 



q 2 l 7 / 5 tanh u 1 tanh 2 u 5 1 \ 

D 2 \256 ~~tf h 256 ~~~u« 25W + 384t*y 



or substituting S = 4u 2 D/l 2 , from Eq. (5), and the expression for Z), 
from Eq. (3), we finally obtain the equation 

E 2 h 8 135 tanh u 27 tanh 2 u _ _135 _9^ 



(1 - v 2 ) 2 q 2 l s 16 u 9 ' 16 u s 16u s ' 8u* 

For a given material, a given ratio /i/Z, and a given load g the left-hand 
side of this equation can be readily calculated, and the value of u satis- 
fying the equation can be found by a trial-and-error method. To simplify 
this solution, the curves shown in Fig. 4 can be used. The abscissas of 
these curves represent the values of u and the ordinates represent the 
quantities logio (10 4 a/C/o), where U denotes the numerical value of the 
right-hand side of Eq. (8). Vf/o is used because it is more easily calcu- 
lated from the plate constants and the load; and the factor 10 4 is intro- 
duced to make the logarithms positive. In each particular case we begin 
by calculating the square root of the left-hand side of Eq. (8), equal to 
£7i 4 /(l - v 2 )ql\ which gives VUo. The quantity log™ (10 4 V~Uo) then 
gives the ordinate which must be used in Fig. 4, and the corresponding 
value of u can be readily obtained from the curve. Having u, we obtain 
the value of the axial force S from Eq. (5). 

In calculating stresses we observe that the total stress at any cross 
section of the strip consists of a bending stress proportional to the bend- 
ing moment and a tensile stress of magnitude S/h which is constant along 
the length of the strip. The maximum stress occurs at the middle of the 
strip, where the bending moment is a maximum. From the differential 
equation (4) the maximum bending moment is 



--*(£). 



=1/2 



1 See Timoshenko, "Strength of Materials," part I, 3d ed., p. 178, 1955. 



BENDING TO A CYLINDRICAL SURFACE 



9 







I 


2 


3 


4 


4 


5 


6 


7 


8 


8 


9 


10 
Value of u 


11 


12 



Fig. 4 
Substituting expression (6) for w, we obtain 

M -& 

- Lr - L ma-: 



8 



fo(u) 



where 



^o = 



1 — sech u 



(9) 
(e) 



The values of ^ are given by curves in Fig. 5. It is seen that these 
values diminish rapidly with increase of u, and for large u the maximum 



(fy^'(^fS>>TUC4(") ^pUD (T1)°J 



2 2 


i 


'OC 

c 




h 

c 


5 


C 


>* c 


3 5 


r N - 
>• c 


) c 


4 — 

>• c 


5 o 


I XDUL 


X 

o 

p 


t/o 

c 
o 

"§ 

0) 

i_ 
c 

H- 

-+- 


cr 
o 

<_> 
a 

a> 

c 
-+- 

O 

.c 
■+- 


00 

C H 

.o 

o 
D 

o> _ 
i_ • 

a> 

"</•> - 

c _ 
a; 

_c - 

-+- ; 


5 

13 
c 
a* 

i_ 

QJ 

c 

O 

n. 

4- 


</0 




p^ 










ending moment= M, 
ef lection = W r 


CD 

O Q> 

o. a 




3 

*+-" 










1 


3 \ 










a. -+- 

I/O CQ 












Mcx.b 
Max.d 


X 

o 

E 


X 

D 

E 


X 

O 

=1 


X 

a 
E 


H 1— 

.9- .9 




-i 
























II 


11 


o <-> 


















*^ 


*. 






00 00 






3 ' 


















^ 








a 


i ^^^ 
































3 
O 






\ 






























































































i£ 



























E 

OO Csl 

t 

E! to 



= -*- 



om 



2 



LO O 



Tf = 



to £i 



~ o 



vo lt\ -*t ro 

o o o o 

cJ <=> c* c> 



10 



BENDING TO A CYLINDRICAL SURFACE II 

bending moment is several times smaller than the moment ql 2 /S which 
would be obtained if there were no tensile reactions at the ends of the 
strip. 

The direct tensile stress cri and the maximum bending stress <j 2 are now 
readily expressed in terms of u, q, and the plate constants as follows : 

S Au 2 D Eu 2 (h 



ai ~ h = ~W~ = 3(1 - v 2 ) (l) (10) 

6 3 / 1\ 2 

* 2 = r2 M max = -g - ^ (11) 



h 2 max 4 y \/i x 
The maximum stress in the plate is then 

0"max = 0"1 + 0"2 

To show how the curves in Figs. 4 and 5 can be used in calculating 
maximum stresses, let us take a numerical example and assume that a 
long rectangular steel plate 50 in. wide and i in. thick carries a uniformly 
distributed load q = 20 psi. We start with a computation of VU~ : 

/Tr E /hV 30 • 10 6 1 „ Mn4a 

^ U «=0^^ q {l) =(T - 0.3*)20 l0* = 0.01648 

Then, from tables, 

log 10 (10 4 VUl) = 2.217 

From the curve A in Fig. 4 we find u = 3.795, and from Fig. 5 we obtain 
^o = 0.1329. 

Now, computing stresses by using Eqs. (10) and (11), we find 

30 - 1Q« - 3.795 2 1 . 

ai = ^_^__^ _ = 15j830 psi 

0-2 = i • 20 • 10 4 • 0.1329 = 19,930 psi 
ffmax = cri + o-2 = 35,760 psi 

In calculating the maximum deflection we substitute x = 1/2 in Eq. (6) 
of the deflection curve. In this manner we obtain 

sech u — 1 + — r 
where f (u) = — ^ 

2T 

To simplify calculations, values of f (u) are given by the curve in Fig. 5. 
If there were no tensile reactions at the ends of the strip, the maximum 



12 



THEORY OF PLATES AND SHELLS 



deflection would be 5ql 4 /3S4D. The effect of the tensile reactions is given 
by the factor fo(u), which diminishes rapidly with increasing u. 

Using Fig. 5 in the numerical example previously discussed, we find 
that for u = 3.795 the value of fo(u) is 0.145. Substituting this value in 
Eq. (12), we obtain 



w n 



= 4.74 • 0.145 = 0.688 in. 



It is seen from Eq. (8) that the tensile parameter u depends, for a 
given material of the plate, upon the intensity of the load q and the 



60,000 



50,000 



.40,000 



! 30,000 



20,000 



10,000 




20 
Load in lb per sq in. 

Fig. 6 



ratio l/h of width to thickness of the plate. From Eqs. (10) and (11) 
we see that the stresses <ri and <r 2 are also functions of u, q, and l/h. 
Therefore, the maximum stress in the plate depends only on the load q 
and the ratio l/h. This means that we can plot a set of curves giving 
maximum stress in terms of q, each curve in the set corresponding to a 
particular value of l/h. Such curves are given in Fig. 6. It is seen that 
because of the presence of tensile forces S, which increase with the load, 
the maximum stress is not proportional to the load q; and for large values 
of q this stress does not vary much with the thickness of the plate. By 
taking the curve marked l/h =100 and assuming q = 20 psi, we obtain 
from the curve the value o- max calculated before in the numerical example. 



BENDING TO A CYLINDRICAL SURFACE 



13 



3. Cylindrical Bending of Uniformly Loaded Rectangular Plates with 
Built-in Edges. We assume that the longitudinal edges of the plate are 
fixed in such a manner that they cannot rotate. Taking an elemental 
strip of unit width in the same manner as before (Fig. 1) and denoting by 
Mo the bending moment per unit length acting on the longitudinal edges 
of the plate, the forces acting on the strip will be as shown in Fig. 7. 
The bending moment at any cross section of the strip is 



M 



ql 



* - %• ~ Sw + Mo 



Substituting this expression in Eq. (4), we obtain 



d 2 w 
dx 2 



S 



qlx qx 2 
2D + 2D 



Mo 
D 



(a) 



The general solution of this equation, using notation (5), will be repre- 
sented in the following form : 



~ . , 2ux . „ , 2ux . ql s x 
w = Cxsmh— + C.cosh^ + ^_ 



ql 2 x 2 



ql* 



+ 



Md 2 



8uW 16u 4 D ' ±u 2 D 



(b) 



Observing that the deflection curve is symmetrical with respect to the 
middle of the strip, we determine the-constants of integration Ci, C 2 , and 



Mo 



TCtmkixixm 



M 



Fig. 7 



the moment M Q from the following three conditions: 



dw 



dx 
w = 



= for x = and x = 



I 



(c) 



for x = 
Substituting expression (6) for w, we obtain from these conditions 

qi 4 n ^ qv 



Ci = 



lQu 3 D 



16u*D 



coth u 



4:U 2 4u 12 v ' 



(13) 



where 



*i(tO 



— %( u ~ tanh u) 
u 2 tanh u 



14 THEORY OF PLATES AND SHELLS 

The deflection w is therefore given by the expression 

ql 4 . , 2ux , ql 4 ,, , 2t&r 

™ = ~ 16^Z> Smh T" + I6^D COth U COsh ~r 

. ql*x ql*x* ql 4 

This can be further simplified and finally put in the following form: 

! cosh \u ( 1 j-j 



qV 

w — 



16m 3 D tanh 7/, I cosh ii 



+ _ 8"^7T~ (14) 



For calculating the parameter u we proceed as in the previous article 
and use Eq. (d) of that article. Substituting in it expression (14) for w 
and performing the integration, we obtain 



5(1- 



■v*)l_q*V(_ 3 1 , 1 , 1 A 

j? D 2 \ 256w 5 tanh u 256m 4 sinh 2 u ~ r 64u« t 384m 4 / 



Substituting <S from Eq. (5) and expression (3) for Z), the equation for 
calculating u finally becomes 

E 2 h* 81. 27 . 27 , 9 „ K . 

+ TTi + cZi (15) 



(1 - v 2 ) 2 qH & 16m 7 tanh w 16m 6 sinh 2 w ' 4m 8 8« 6 

To simplify the solution of this equation we use the curve in Fig. 8, in 
which the parameter u is taken as abscissa and the ordinates are equal 
to logio (10 4 \/U\) ,where U\ denotes the right-hand side of Eq. (15). 
For any given plate we begin by calculating the square root of the left- 
hand side of Eq. (15), equal_to Eh*/[(1 — v*)ql 4 ], which gives us Wi. 
The quantity logio (10 4 s/Ui) then gives the ordinate of the curve in 
Fig. 8, and the corresponding abscissa gives the required value of u. 

Having u, we can begin calculating the maximum stresses in the plate. 
The total stress at any point of a cross section of the strip consists of the 
constant tensile stress <x\ and the bending stress. The maximum bending 
stress ff 2 will act at the built-in edges where the bending moment is the 
largest. Using Eq. (10) to calculate <j x and Eq. (13) to calculate the 
bending moment Mo, we obtain 

Eu 2 (h\- , 1A v 

= 3- ( T^Mr) (16) 

0"max = Ori + CT2 

To simplify the calculation of the stress a 2 , the values of the function 
yj/\{u) are given by a curve in Fig. 5. 

The maximum deflection is at the middle of the strip and is obtained by 



<ri 



BENDING TO A CYLINDRICAL SURFACE 



15 



13i 1 4.0 




OJ-Ll.CH-I.O- 






2 


3 


4 


A- ! 


3 6 


7 


8 


8 < 


J l0 . 

Value of u 

Fig. 8 


II 


12 



substituting x = 1/2 in Eq. (14), from which 



qt 4 * / \ 



where 



/iC 



. = 24 /m_ 2 ^ u \ 

W ' ~ it 4 \ 2 sinh w tanh u) 



2 sinh w 
The function fi(u) is also given by a curve in Fig. 5. 



(is: 



16 



THEORY OF PLATES AND SHELLS 



The use of the curves in Figs. 5 and 8 will now be illustrated 
numerical example. A long rectangular steel plate has the dimei 
I = 50 in., h = | in., and q = 10 psi. In such a case we have 



VUi = 



E 



/h\ = 30- 

\l) (1 - 0.3 2 



10 6 



(1 - v*)q\lj (1 - 0.3 2 )10- 10 4 
log 10 10 4 VUl = 2.5181 



= 0.032966 



From Fig. 8 we now lind u = 1.894; and from Fig. 5, ^i - 0.8212. 
stituting these values in Eqs. (16) and (17), we find 



<7l = 



30 • 10 6 • 1.894 2 



3(1 - 0.3 2 )10 4 
a t = i ■ 10 • 10 4 • 0.8212 = 
ffmax = ai + a 2 — 45,000 psi 



= 3,940 psi 

41,060 psi 



Comparing these stress values with the maximum stresses obtaine 
a plate of the same size, but with twice the load, on the assumpti 



60,000 



50,000 



. 40,000 



CD 
Ol 



£ 30,000 



20,000 



10,000 




6 7 8 9 10 
Lood in lb per sq in. 

Fig. 9 



14. 15 



simply supported edges (see page 11), it can be concluded that, owii 
clamping of the edges, the direct tensile stress decreases consider* 
whereas the maximum bending stress increases several times, so 
finally the maximum total stress in the case of clamped edges bec< 
\arger than in the case of simply supported edges. 



BENDING TO A CYLINDRICAL SURFACE 17 

Proceeding as in the previous article it can be shown that the maxi- 
mum stress in a plate depends only on the load q and the ratio l/h, and 
we can plot a set of curves giving maximum stress in terms of q, each 
curve in the set corresponding to a particular value of l/h. Such curves 
are given in Fig. 9. It is seen that for small values of the intensity of 
the load q, when the effect of the axial force on the deflections of the 
strip is small, the maximum stress increases approximately in the same 
ratio as q increases. But for larger values of q the relation between the 
load and the maximum stress becomes nonlinear. 

In conclusion, we give in Table 1 the numerical values of all the func- 
tions plotted in Figs. 4, 5, and 8. This table can be used instead of the 
curves in calculating maximum stresses and maximum deflections of long, 
uniformly loaded rectangular plates. 

4. Cylindrical Bending of Uniformly Loaded Rectangular Plates with 
Elastically Built-in Edges. Let us assume that when bending occurs, 
the longitudinal edges of the plate rotate through an angle proportional 
to the bending moment at the edges. In such a case the forces acting on 
an elemental strip will again be of the type shown in Fig. 7, and we shall 
obtain expression (6) of the previous article for the deflections w. How- 
ever, the conditions at the edges, from which the constants of integration 
and the moment M are determined, are different; viz., the slope of the 
deflection curve at the ends of the strip is no longer zero but is propor- 
tional to the magnitude of the moment M , and we have 

P) = — iSAfo (o) 

ax /x=o 

where p is a factor depending on the rigidity of restraint along the edges. 
If this restraint is very flexible, the quantity j8 is large, and the conditions 
at the edges approach those of simply supported edges. If the restraint 
is very rigid, the quantity becomes small, and the edge conditions 
approach those of absolutely built-in edges. The remaining two end 
conditions are the same as in the previous article. Thus we have 

(w) x -o = 

Using these conditions, we find both the constants of integration and the 
magnitude of M in expression (b) of the previous article. Owing to 
flexibility of the boundary, the end moments M will be smaller than 
those given by Eq. (13) for absolutely built-in edges, and the final result 
can be put in the form 

Mo=-t^.(«) a 9 ) 



(: 



18 



THEORY OF PLATES AND SHELLS 
Table 1 





















\ 




u 


log 10 10 4 


VUo 


log.o 10 4 


y/Ui 


log 10 10 4 


VUt 


/o(«) 


/.(«) 


^o(w) 


*l(«) 


u 





00 




OO 




OO 




1.000 


1.000 


1.000 


1.000 





0.5 


3.889 


406 


3.217 


331 


3.801 


425 


0.908 


0.976 


0.905 


0.984 


0.5 


1.0 


3.483 


310 


2.886 


223 


3.376 


336 


0.711 


0.909 


0.704 


0.939 


1.0 


1.5 


3.173 


262 


2.663 


182 


3.040 


292 


0.532 


0.817 


0.511 


0.876 


1.5 


2.0 


2.911 


227 


2.481 


161 


2.748 


257 


0.380 


0.715 


0.367 


0.806 


2.0 


2.5 


2.684 


198 


2.320 


146 


2.491 


228 


0.281 


0.617 


0.268 


0.736 


2.5 


3.0 


2.486 


175 


2.174 


134 


2.263 


202 


0.213 


0.529 


0.200 


0.672 


3.0 


3.5 


2.311 


156 


2.040 


124 


2.061 


180 


0.166 


0.453 


0.153 


0.614 


3.5 


4.0 


2.155 


141 


1.916 


115 


1.881 


163 


0.132 


0.388 


0.120 


0.563 


4.0 


4.5 


2.014 


128 


1.801 


107 


1.718 


148 


0.107 


0.335 


0.097 


0.519 


4.5 


5.0 


1.886 


118 


1.694 


100 


1.570 


135 


0.088 


0.291 


0.079 


0.480 


5.0 


5.5 


1.768 


108 


1.594 


93 


1.435 


124 


0.074 


0.254 


0.066 


0.446 


5.5 


6.0 


1.660 


100 


1.501 


88 


1.311 


115 


0.063 


0.223 


0.055 


0.417 


6.0 


6.5 


1.560 


93 


1.413 


82 


1.196 


107 


0.054 


0.197 


0.047 


0.391 


6.5 


7.0 


1.467 


87 


1.331 


78 


1.089 


100 


0.047 


0.175 


0.041 


0.367 


7.0 


7.5 


1.380 


82 


1.253 


74 


0.989 


94 


0.041 


0.156 


0.036 


0.347 


7.5 


8.0 


1.298 


77 


1 .179 


70 


0.895 


89 


0.036 


0.141 


0.031 


0.328 


8.0 


8.5 


1.221 


73 


1.109 


67 


0.806 


83 


0.032 


0.127 


0.028 


0.311 


8.5 


9.0 


1.148 


69 


1.042 


63 


0.723 


80 


0.029 


0.115 


0.025 


0.296 


9.0 


9.5 


1.079 


65 


0.979 


61 


0.643 


75 


0.026 


0.105 


0.022 


0.283 


9.5 


10.0 


1.014 


63 


0.918 


58 


0.568 


72 


0.024 


0.096 


0.020 


0.270 


10.0 


10.5 


0.951 


59 


0.860 


55 


0.496 


69 


0.021 


0.088 


0.018 


0.259 


10.5 


11.0 


0.892 


57 


0.805 


54 


0.427 


65 


0.020 


0.081 


0.017 


0.248 


11.0 


11.5 


0.835 


55 


0.751 


51 


0.362 


63 


0.018 


0.075 


0.015 


0.238 


11.5 


12.0 


0.780 




0.700 




0.299 




0.016 


0.069 


0.014 


0.229 


12.0 



BENDING TO A CYLINDRICAL SURFACE 19 

where 7 is a numerical factor smaller than unity and given by the formula 

__ tanh u 

^ ~ 28 

— Du + tanh u 

It is seen that the magnitude of the moments M at the edges depends 
upon the magnitude of the coefficient /3 defining the rigidity of the 
restraint. When is very small, the coefficient 7 approaches unity, 
and the moment Mo approaches the value (13) calculated for absolutely 
built-in edges. When /3 is very large, the coefficient 7 and the moment 
Mo become small, and the edge conditions approach those of simply 
supported edges. 

The deflection curve in the case under consideration can be repre- 
sented in the following form : 



w 



ql 4 tanh u — 7 (tanh u — u) j L v „ f j _ . 



|cosh[ W (l-^)J 



lQu 4 D tanh u \ cosh u 



+ Sd*«- x) < 20 > 

For 7=1 this expression reduces to expression (14) for deflections of a 
plate with absolutely built-in edges. For 7 = we obtain expression (6) 
for a plate with simply supported edges. 

In calculating the tensile parameter u we proceed as in the previous 
cases and determine the tensile force S from the condition that the exten- 
sion of the elemental strip is equal to the difference between the length of 
the arc along the deflection curve and the chord length /. Hence 

3(1 - v 2 )l _ 1 f l fdw\ 2 



hE 



= 1 n (dw\ 



Substituting expression (20) in this equation and performing the inte- 
gration, we obtain 

EW (1 -y)Uo + yUi - 7(1 - y)U* (21) 



(1 - v 2 ) 2 qH* 

where C7 and Uj denote the right-hand sides of Eqs. (8) and (15), respec- 
tively, and 

rr 27 (u — tanh u) 2 , . , 9 . x u \ 

U * = 16 i.'tanh»« (M tanh U ~ U + tanh U) 

The values of logio (10 4 VVl) are given in Table 1. By using this table, 
Eq. (21) can be readily solved by the trial-and-error method. For any 
particular plate we first calculate the left-hand side of the equation and, 



20 THEORY OF PLATES AND SHELLS 

by using the curves in Figs. 4 and 8, determine the values of the parame- 
ter u (1) for simply supported edges and (2) for absolutely built-in edges. 
Naturally u for elastically built-in edges must have a value intermediate 
between these two. Assuming one such value for u, we calculate £/ , J7i, 
and U 2 by using Table 1 and determine the value of the right-hand side 
of Eq. (21). Generally this value will be different from the value of the 
left-hand side calculated previously, and a new trial calculation with a 
new assumed value for u must be made. Two such trial calculations 
will usually be sufficient to determine by interpolation the value of u 
satisfying Eq. (21). As soon as the parameter u is determined, the bend- 
ing moments M at the ends may be calculated from Eq. (19). We can 
also calculate the moment at the middle of the strip and find the maxi- 
mum stress. This stress will occur at the ends or at the middle, depend- 
ing on the degree of rigidity of the constraints at the edges. 

5. The Effect on Stresses and Deflections of Small Displacements of 
Longitudinal Edges in the Plane of the Plate. It was assumed in the 
previous discussion that, during bending, the longitudinal edges of the 
plate have no displacement in the plane of the plate. On the basis of this 
assumption the tensile force S was calculated in each particular case. 
Let us assume now that the edges of the plate undergo a displacement 
toward each other specified by A. Owing to this displacement the 
extension of the elemental strip will be diminished by the same amount, 
and the equation for calculating the tensile force S becomes 

Sl(l - v*) 1 P (dw\ , A , . 

KE = 2JoV^J **-* (a) 

At the same time Eqs. (6), (14), and (20) for the deflection curve hold 
true regardless of the magnitude of the tensile force S. They may be 
differentiated and substituted under the integral sign in Eq. (a). After 
evaluating this integral and substituting S = 4u 2 D/l 2 , we obtain for 
simply supported edges 



E*h* u2 + 



3ZA 



q 2 (l - v 2 ) 2 l* u 2 



= U (22) 



and for built-in edges 



2 j_ ^ 
E*h* U "*" h 2 



q 2 (l - v 2 )H* u 2 Ul (23) 

If A is made zero, Eqs. (22) and (23) reduce to Eqs. (8) and (15), obtained 
previously for immovable edges. 

The simplest case is obtained by placing compression bars between the 
longitudinal sides of the boundary to prevent free motion of one edge of 



BENDING TO A CYLINDRICAL SURFACE 



21 



the plate toward the other during bending. Tensile forces S in the plate 
produce contraction of these bars, which results in a displacement A pro- 
portional to S.* If fc is the factor of proportionality depending on the 
elasticity and cross-sectional area of the bars, we obtain 

S = kA 

or, substituting S = 4:U 2 D/l 2 , we obtain 

1 Eu 2 h z 



k 3Z 2 (1 - v 2 ) 



u 2 + 



3ZA 



and 



u l 



= 1 + 



Eh 



kl(l - v 2 ) 



Thus the second factor on the left-hand side of Eqs. (22) and (23) is a 
constant that can be readily calculated if the dimensions and the elastic 
properties of the structure are known. Having the magnitude of this 
factor, the solution of Eqs. (22) and (23) can be accomplished in exactly 
the same manner as used for immovable edges. 




ro m t 



n Pj 



Fig. 10 



In the general case the second factor on the left-hand side of Eqs. (22) 
and (23) may depend on the magnitude of the load acting on the struc- 
ture, and the determination of the parameter u can be accomplished only 
by the trial-and-error method. This procedure will now be illustrated 
by an example that is encountered in analyzing stresses in the hull of a 
ship when it meets a wave. The bottom plates in the hull of a ship are 
subjected to a uniformly distributed water pressure and also to forces in 
the plane of the plates due to bending of the hull as a beam. Let 6 be 
the width of the ship at a cross section mn (Fig. 10) and I be the frame 
spacing at the bottom. When the hollow of a wave is amidships (Fig. 
116), the buoyancy is decreased there and increased at the ends. The 
effect of this change on the structure is that a sagging bending moment 
is produced and the normal distance I between the frames at the bottom 
is increased by a certain amount. To calculate this displacement accu- 
rately we must consider not only the action of the bending moment M 
on the hull but also the effect on this bending of a certain change in 

* The edge support is assumed to be such that A is uniform along the edges. 



22 



THEORY OF PLATES AND SHELLS 






Sagging 

(b) 
Fig. 11 

tensile forces S distributed along the edges ran and ra x ni of the bottom 
plate mnmiUi (Fig. 10), which will be considered as a long rectangular 
plate uniformly loaded by water pressure. Owing to the fact that the 

plates between the consecutive 



WWl 



Sb 
o) 



frames are equally loaded, there 
will be no rotation at the longitu- 
dinal edges of the plates, and they 
may be considered as absolutely 
built in along these edges. 

To determine the value of A, 
which denotes, as before, the dis- 
placement of the edge ran toward 
the edge raini in Fig. 10 and which 
is produced by the hull bending 
moment M and the tensile reactions 
S per unit length along the edges 
ran and raini of the bottom plate, let 
us imagine that the plate ranra x ni is 
removed and replaced by uniformly 
distributed forces S so that the to- 
tal force along ran and ra^i is Sb 
(Fig. 12a). We can then say that 
the displacement A of one frame 
relative to another is due to the 
bending moment M and to the 

eccentric load Sb applied to the hull without bottom plating. 

If A, 7, and c are the cross-sectional area, the centroidal moment of 

inertia, and the distance from the bottom plate to the neutral axis of the 




Centroid A t 



Centrofd A 



PI 



A 



h — V 



J. JL- 



|<- b _.. 

(b) 
Fig. 12 



BENDING TO A CYLINDRICAL SURFACE 23 

complete hull section, and if A\, 7i, and Ci are the corresponding quanti- 
ties for the hull section without bottom plates, the latter set of quantities 
can be derived from the former by the relations 



A x = A - bh 

- A£ 
01 A, 

h = I - bhc* - A^d - c) 2 



(b) 



The relative displacement Ai produced by the eccentrically applied forces 
Sb is 

Id ~ 



A, = 



jS) (Sb Sbc{ \ 



E 

in which the factor 1 — v 2 must be introduced if one neglects the lateral 
strain. The displacement due to the bending moment M is 

Mai 

A2= -^77 

Hence the total displacement is 



A = A x + A 2 = Z(1 



E 
Substituting in this expression 

4u 2 D Eu 2 h 



v 2 ) r sb , sbci id i , N 
-[a7 + -/7-j 1 (i-^)J (c) 



s = 



Z 2 3Z 2 (1 - v*) 

we finally obtain 

A u% s ( b . bc{\ Mia , « 

A = ^T U + 77J " ^77 (d) 

This quantity must be substituted in Eq. (23) for determining the tensile 
parameter u. 

Let us apply this theory to a numerical example. Assume b — 54 ft, 
I = 1,668 ft 4 , A = 13.5 ft 2 , c = 12.87 ft, h = 0.75 in. = 0.0625 ft, 
I = 45 in. = 3.75 ft, q = 10 psi, M = 123,500 ft-tons. From Eqs. (6) 
we obtain 

Ai = 13.5 - 0.0625 • 54 = 10.125 ft 2 

13.5 -12.87 171ftf . 
Cl= 10.125 =1716ft 
h = 1,668 - 559.0 - 10.125(17.16 - 12.87) 2 = 922.7 ft 4 

Substituting these values in expression (d), we calculate A and finally 
obtain 

^ = 1.410m 2 - 11.48 
h 2 



24 THEORY OF PLATES AND SHELLS 

Equation (23) then becomes 

E 2 h* u 2 + lAlOu 2 - 11.48 



or 



q 2 (l ~ v 2 ) 2 l s u 2 

1.552M 4 ju 2 - 4.1 
g(l - *> 2 )/ 4 \ u 2 



- ffi 



Substituting numerical values and taking logarithms of both sides, 
we obtain 



3.597 + logio ^ J' 763 = logic (10 4 VUl) 

Using the curve in Fig. 8, this equation can be readily solved by the 
trial-and-error method, and we obtain u = 2.187 and, from Fig. 5, 
^i(w) = 0.780. The maximum stress is now calculated by using Eas. 
(16) and (17), from which 

30 ; 10* ; 4.783 1AA „ . 
ai = 3 - 0.91 - 60* = 14 ' 600 PS1 
a 2 = i • 10 • 60 2 • 0.780 = 14,040 psi 

<Tma X = <Tl + <T2 = 28,640 psi 

If the bending stress in the plate due to water pressure were neglected 
and if the bottom plate stress were calculated from the formula a = Mc/I, 
we would arrive at a figure of only 13,240 psi. 

6. An Approximate Method of Calculating the Parameter u. In calcu- 
lating the parameter u for plates in which the longitudinal edges do not 
move in the plane of the plate, we used the equation 



which states that the extension of an elemental strip produced by the 
forces S is equal to the difference between the length of the arc along the 
deflection curve of the strip and the chord length I. In the particular 
cases considered in the previous articles, exact expressions for the deflec- 
tions w were derived, and numerical tables and curves for the right-hand 
side of Eq. (a) were given. When such tables are not at hand, the solu- 
tion of the equation becomes complicated, and to simplify the problem 
recourse should be had to an approximate method. From the discussion 
of bending of beams it is known 1 that, in the case of simply supported 
ends with all lateral loads acting in the same direction, the deflection 
curve of an elemental strip produced by a combination of a lateral load 
and an axial tensile force S (Fig. 3) can be represented with sufficient 
1 See Timoshenko, " Strength of Materials," part II. 3d ed., p. 52, 1956. 



BENDING TO A CYLINDRICAL SURFACE 25 

accuracy by the equation 

w = 



Wo • TX /7x 

sin -7- (o; 



1 + a I 

in which w denotes the deflection at the middle of the strip produced by 
the lateral load alone, and the quantity a is given by the equation 

S SI 2 , . 

Thus, a represents the ratio of the axial force S to the Euler critical load 
for the elemental strip. 

Substituting expression (6) in Eq. (a) and integrating, we obtain 



Sl(l - V 2 ) _ 1T 2 W 







hE 4/(1 -f a) 2 

Isiow, using notation (c) and substituting for D its expression (3), we 
finally obtain 

«(1 + a) 2 = ^° (24) 

From this equation the quantity a can be calculated in each particular 
case, and the parameter u is now determined from the equation 



u 2 = 



D 4 4 



(d) 



To show the application of the approximate Eq. (24) let us take a 
numerical example. A long rectangular steel plate with simply sup- 
ported edges and of dimensions I = 50 in. and h = % in. is loaded with a 
uniformly distributed load q = 20 psi. In such a case 

5 Q]l 
Wq "" 384 D 

and, after substituting numerical values, Eq. (24) becomes 

a(l + a) 2 = 269.56 

The solution of the equation can be simplified by letting 

1 + a = x (e) 

Then a; 3 - x 2 = 269.56 

i.e., the quantity x is such that the difference between its cube and its 
square has a known value. Thus x can be readily determined from a 
slide rule or a suitable table, and we find in our case 

x = 6.8109 and a = 5.8109 



26 THEOin OF PLATES AND SHELLS 

Then, from Eq. (d) 

u = 3.7865 

and from the formula (e) (see page 9) 

^o = 0.13316 

For calculating direct stress and maximum bending stress we use Eqs. 
(10) and (11). In this way we find 

cri = 15,759 psi 
(7 2 = 19,974 psi 
o"max = 0-1 + 0-2 = 35,733 psi 

The calculations made in Art. 2 (page 11) give, for this example, 

0- m ax = 35,760 psi 

Thus the accuracy of the approximate Eq. (24) is in this case very high. 
In general, this accuracy depends on the magnitude of u. The error 
increases with increase of u. Calculations show that for u = 1.44 the 
error in the maximum stress is only 0.065 of 1 per cent and that for 
u = 12.29, which corresponds to very flexible plates, it is about 0.30 of 
1 per cent. These values of u will cover the range ordinarily encountered 
in practice, and we conclude that Eq. (24) can be used with sufficient 
accuracy in all practical cases of uniformly loaded plates with simply 
supported edges. 

It can also be used when the load is not uniformly distributed, as in 
the case of a hydrostatic pressure nonuniformly distributed along the 
elemental strip. If the longitudinal force is found by using the approxi- 
mate Eq. (24), the deflections may be obtained from Eq. (6), and the 
bending moment at any cross section may be found as the algebraic sum 
of the moment produced by the lateral load and the moment due to the 
longitudinal force. 1 

In the case of built-in edges the approximate expression for the deflec- 
tion curve of an elemental strip can be taken in the form 

Wo 1 /-, 2irx\ /Jr . 

in which Wo is the deflection of the built-in beam under the lateral load 
acting alone and a has the same meaning as before. Substituting this 
expression in Eq. (a) and integrating, we obtain for determining a the 
equation 

1 More accurate values for the deflections and for the bending moments can be 
obtained by substituting the approximate value of the longitudinal force in Eq. (4) 
and integrating this equation, which gives Eqs. (12) and (9). 



BENDING TO A CYLINDRICAL SURFACE 27 

which can be solved in each particular case by the method suggested for 
solving Eq. (24). 

When a is found, the parameter u is determined from Eq. (d); the 
maximum stress can be calculated by using Eqs. (16) and (17); and the 
maximum deflection, by using Eq. (18). 

If, during bending, one edge moves toward the other by an amount A, 
the equation 

sia - s) _ i r (dwy dx _ A to) 



hE 2 Jo \dx ) 

must be used instead of Eq. (a). Substituting expression (6) in this 
equation, we obtain for determining a in the case of simply supported 
edges the equation 

a + 12 — - 

a{l + a) 1 = --TT- Uo; 

In the case of built-in edges we use expression (/). Then for determin- 
ing a we obtain 



-( i+ s)' 



4- 19 A/ 



_ Z 2 ^ = ^ 



(27) 



If the dimensions of the plate and the load q are given, and the displace- 
ment A is known, Eqs. (26) and (27) can both be readily solved in the 
same manner as before. If the displacement A is proportional to the 
tensile force S, the second factor on the left-hand sides of Eqs. (26) and 
(27) is a constant and can be determined as explained in the previous 
article (see page 21). Thus again the equations can be readily solved. 

7. Long Uniformly Loaded Rectangular Plates Having a Small Initial 
Cylindrical Curvature. It is seen from the discussions in Arts. 2 and 3 
that the tensile forces S contribute to the strength of the plates by 
counteracting the bending produced by lateral load. This action 
increases with an increase in deflection. A further reduction of maxi- 
mum stress can be accomplished by giving a suitable initial curvature 
to a plate. The effect on stresses and deflections of such an initial curva- 
ture can be investigated 1 by using the approximate method developed in 
the previous article. 

Let us consider the case of a long rectangular plate with simply sup- 
ported edges (Fig. 13), the initial curvature of which is given by the 
equation 

3 See S. Timoshenko's paper in "Festschrift zum siebzigsten Geburtstagc August 
Foppls," p. 74, Berlin, 1923. 



28 



THEORY OF PLATES AND SHELLS 



Wi = 5 sin 



TX 



I 



(a) 



If tensile forces S are applied to the edges of the plate, the initial 
deflections (a) will be reduced in the ratio 1/(1 + a), where a has the 
same meaning as in the previous article 1 (page 25). The lateral load in 
combination with the forces S will produce deflections that can be 
expressed approximately by Eq. (6) of the previous article. Thus the 
total deflection of the plate, indicated in Fig. 13 by the dashed line, is 



w 



o . TTX Wo . TTX 

1 + a I 1 -\- a I 



8 + Wo . TTX 

T~- L — sin-y- 
1 + a I 



(b) 



Assuming that the longitudinal edges of the plate do not move in the 
plane of the plate, the tensile force S is found from the condition that 
the extension of the elemental strip produced by the forces S is equal to 



s 


"TTTT-r-rT Z -r-rrrTi 


- s 




^Mi-4. ■■& .£ JsM"- 


[ql 
2 


W ^W| 



Fig. 13 



the difference between the length of the arc along the deflection curve 
of the elemental strip and the initial length of the strip. This difference, 
in the case of small deflections, is given by the equation 



i-im*- 1 *!:®)'* 



(c) 



Substituting expressions (a) and (6) for w and Wi and integrating, we 
obtain 



»-S[(^5)"-'] 



Putting X equal to the extension of the strip Sl(l — v 2 )/hE, we finally 
obtain 



n 4- ^2 _ 3(5 + wo)* 33 2 (1 + ccY 
{1 + a) IF h* 



(28) 



If we take 6 = 0, this equation reduces to Eq. (24) for a plate without 
initial curvature. 

To show the effect of the initial curvature on the maximum stress in a 
plate, let us apply Eq. (28) to a numerical example. Assume that a 
steel plate having I = 45 in. and h = f in. is submitted to the action of 

1 See Timoshenko, "Strength of Materials," part II, 3d ed., p. 52, 1956. 



BENDING TO A CYLINDRICAL SURFACE 29 

ft uniformly distributed load q = 10 psi. If there is no initial deflection, 
5 = and Eq. (28) becomes 

o(l + a) 2 = 290 
from which 

a = 5.97 and u = o ^^ ~ 3 *^ 

From Eq. (10) we then obtain 

<n = 11,300 psi 

and from Eq. (11) 

at = 14,200 psi 

The maximum stress in the plate is 

<w = <ri + 0-2 = 25,500 psi 

Let us now assume that there is an initial deflection in the plate such that 
5 = h = f in. In such a case Eq. (28) gives 

a(l + a) 2 = 351.6 - 3(1 + a) 2 

Letting 

1 + a = X 

we obtain 

x s _|_ 2x2 = 351 . 6 

from which 

x = 6.45 a = 5.45 u = | V^ = 3.67 

The tensile stress, from Eq. (10), is 

en = 10,200 psi 

In calculating the bending stress we must consider only the change in 
deflections 

Wo ■ irx ac8 . tx /« 

w - W! = ^-f— sin -j- - r—r— sin -y- (rf) 

1 + a / 1 + a t 

The maximum bending stress, corresponding to the first term on the 
right-hand side of Eq. (d), is the same as for a flat plate with u = 3.67. 
From Table 1 we find rpo = 0.142 and from Eq. (11) 

<r' 2 = 15,300 psi 

The bending moment corresponding to the second term in Eq. (d) is 

_ d* / ad . ttx\ OTr z 8D . wX 

- D d ^{-rT^ sm T) == -(TTajp sm T 



30 



THEORY OF PLATES AND SHELLS 



This moment has a negative sign, and a corresponding maximum stress of 



<$ 



air' 



bD 



6 



h 2 (1 + a)P 



= 9,500 psi 



must be subtracted from the bending stress <t' 2 calculated above. Hence 
the maximum stress for the plate with the initial deflection is 

<w = 10,200 + 15,300 - 9,500 = 16,000 psi 

Comparison of this result with that obtained for the plane plate shows 
that the effect of the initial curvature is to reduce the maximum stress 
from 25,500 to 16,000 psi. This result is obtained assuming the initial 
deflection equal to the thickness of the plate. By increasing the initial 
deflection, the maximum stress can be reduced still further. 

8. Cylindrical Bending of a Plate on an Elastic Foundation. Let us consider the 
problem of bending of a long uniformly loaded rectangular plate supported over the 
entire surface by an elastic foundation and rigidly supported along the edges (Fig. 14). 




Cutting out from the plate an elemental strip, as before, we may consider it as a beam 

on an elastic foundation. Assuming that the reaction of the foundation at any point 

is proportional to the deflection w at that point, and using Eq. (4), we obtain by double 

differentiation of that equation 1 

d 4 w 
D-— = q - Jew (29) 

dx 4 

where q is the intensity of the load acting on the plate and k is the reaction of the 
foundation per unit area for a deflection equal to unity. Introducing the notation 



(30) 




the general solution of Eq. (29) can be written as follows: 

q „ . 2px . , 20x , „ . 2/3* 2f3x 2f3x . 2/3* 

w = - + Ci sin — smh — - + C 2 sin — cosh — + C 3 cos — sinh — 
k 11 It ci 

2px 2px 

+ C t cos — cosh — — 

L I 



(a) 



The four constants of integration must now be determined from the conditions at the 
ends of the strip. In the case under consideration the deflection is symmetrical with 
respect to the middle of the strip. Thus, taking the coordinate axes as shown in Fig. 

i Ibid., p. 21. 



BENDING TO A CYLINDRICAL SURFACE 31 

14, we conclude 1 that C 2 = C 3 = 0. The constants d and C 4 are found from the 
conditions that the deflection and the bending moment of the strip are zero at the end 
(x = 1/2). Hence 

(w)x=m = 



(dhv\ 
dx 2 Jx-l/2 



(b) 

Substituting expression (a) for w and observing that <7 2 = Ca = 0, we obtain 

- + Ci sin sinh + C 4 cos cosh = ( 

k \ c ) 

Ci cos cosh — C 4 sin sinh 0=0 

from which we find 

g sin sinh ff 2 sin sinh 



Ci = -7 



*--? 



Jfc sin 2 sinh 2 + cos 2 cosh 2 4 cos 20 + cosh 20 

cos cosh g 2 cos cosh 



/c sin 2 sinh 2 + cos 2 cosh 2 & cos 20 + cosh 20 

Substituting these values of the constants in expression (a) and using Eq. (30), we 
finally represent the deflection of the strip by the equation 



ql 4 / 2 sin sinh . 20* . , 26x 

I 1 — — - sin — - sinh —— 

1D0 4 \ cos 20 + cosh 20 I I 

) 



2 cos cosh 20s , 20x\ 

— — — cos -— cosh — - J (d) 

cos 20 + cosh 20 I I 

The deflection at the middle is obtained by substituting x = 0, which gives 

<">■-- S3 •<» <31) 

6 / 2 cos ft cosh j8 \ 

where *<*) 'wy- cos 20 + cosh 2fi) 

To obtain the angles of rotation of the edges of the plate, we differentiate expression 
(d) with respect to x and put x = — 1/2. In this way we obtain 

(*»\ - JH ^(0) (32) 

3 sinh 20 - sin 20 

wri ere <pi(0) = tz: r 0i0 ; ^T" 

40 3 cosh 20 + cos 20 

The bending moment at any cross section of the strip is obtained from the equation 

r,d 2 w 

M = -D — 

da: 2 

Substituting expression (d) for w, we find for the middle of the strip 

(ilf ),-o = ? * 2 (0) (33) 

o 

2 sinh sin 
where ^2(0) = 



2 cosh 20 + cos 20 

1 It is seen that the terms with coefficients C 2 and C 3 change sign when x is replaced 
by -x. 



32 



THEORY OF PLATES AND SHELLS 



To simplify the calculation of deflections and stresses, numerical values of functions 
<p, <pi, and ip2 are given in Table 2. For small values of p, that is, for a yielding founda- 
tion, the functions <p and <P2 do not differ greatly from unity. Thus the maximum 
deflection and bending stresses are close to those for a simply supported strip without 
an elastic foundation. With an increase in 0, the effect of the foundation becomes 
more and more important. 



1 


p z p 

1 I 

2 2 ., * 




w/^/gw^y^/fc ?sw^/^^/^/^ 


s 






w 


* 



Fig. 15 

Conditions similar to those represented in Fig. 14 are obtained if a long rectangular 
plate of width I is pressed into an elastic foundation by loads uniformly distributed 
along the edges and of the amount P per unit length (Fig. 15). The plate will be 

Table 2 






<P 


<P\ 


<P2 


fi 


<p 


Vl 


<P2 


0.1 


1.000 


1.000 


1.000 


1.6 


0.186 


0.200 


0.164 


0.2 


0.999 


0.999 


0.999 


1.7 


0.151 


0.166 


0.129 


0.3 


0.996 


0.995 


0.995 


1.8 


0.124 


0.138 


0.101 


0.4 


0.984 


0.983 


0.983 


1.9 


0.102 


0.116 


0.079 


0.5 


0.961 


0.961 


0.959 


2.0 


0.084 


0.099 


0.062 


0.6 


0.921 


0.923 


0.919 


2.2 


0.058 


0.072 


0.037 


0.7 


0.863 


0.866 


0.859 


2.4 


0.042 


0.055 


0.021 


0.8 


0.787 


0.791 


0.781 


2.6 


0.029 


0.043 


0.011 


0.9 


0.698 


0.702 


0.689 


2.8 


0.022 


0.034 


0.005 


1.0 


0.602 


0.609 


0.591 


3.0 


0.016 


0.028 


0.002 


1.1 


0.508 


0.517 


0.494 


3.2 


0.012 


0.023 


0.000 


1.2 


0.421 


0.431 


0.405 


3.4 


0.010 


0.019 


-0.001 


1.3 


0.345 


0.357 


0.327 


3.6 


0.007 


0.016 


-0.002 


1.4 


0.281 


0.294 


0.262 


3.8 


0.006 


0.014 


-0.002 


1.5 


0.228 


0.242 


0.208 


4.0 


0.005 


0.012 


-0.002 



pressed into the elastic foundation and bent, as shown by the dashed line. If 8 denotes 
the deflection at the edges of the plate, the reaction of the foundation at any point is 

k(8 — w) = k8 — kw 

where w is given by Eq. (d) with q = k8. The magnitude of 8 is then obtained from 
the condition that the load is balanced by the reaction of the foundation. Hence 



P = 



kM 

T 



[1/2 

— k I w dx 
Jo 



Plates on elastic foundation with other conditions at the longitudinal edges can 
also be discussed in a similar manner. 



CHAPTER 2 



PURE BENDING OF PLATES 



9. Slope and Curvature of Slightly Bent Plates. In discussing small 
deflections of a plate we take the middle plane of the plate, before bend- 
ing occurs, as the xy plane. During bending, the particles that were in 
the xy plane undergo small displacements w perpendicular to the xy plane 
and form the middle surface of the plate. These displacements of the 
middle surface are called deflections of a plate in our further discussion. 
Taking a normal section of the plate parallel 
to the'xz plane (Fig. 16a), we find that the 
slope of the middle surface in the x direction 
is i x = dw/dx. In the same manner the slope 
in the y direction is i y = dw/dy. Taking 
now any direction an in the xy plane (Fig. 
166) making an angle a with the x axis, we find 
that the difference in the deflections of the two 
adjacent points a and a\ in the an direction is 




m — 



7 dw 1 . to, 

dw = — dx + t— dy 
dx dy 

and that the corresponding slope is 

dw _ dw dx , dw dy 
dn dx dn dy dn 



a 


^ x * 




yf 


\dn 


d 


f 


a 




,N ^n 



Fig. 16 



dw , dw . 

-r— cos a + — sin a 
arc dy 



(a) 



To find the direction a\ for which the slope is a maximum we equate to 
zero the derivative with respect to a of expression (a). In this way we 
obtain 



tan 



dw /dw 



(b) 



Substituting the corresponding values of sin ai and cos «i in (a), we obtain 
for the maximum slope the expression 






(c) 



By setting expression (a) equal to zero we obtain the direction for which 

33 



34 THEORY OF PLATES AND SHELLS 

the slope of the surface is zero. The corresponding angle a 2 is deter- 
mined from the equation 

dw I dw / 7N 

tan „,= -_/_ (d) 

From Eqs. (6) and (d) we conclude that 

tan a\ tan a 2 = — 1 

which shows that the directions of zero slope and of maximum slope are 
perpendicular to each other. 

In determining the curvature of the middle surface of the plate we 
observe that the deflections of the plate are very small. In such a case 
the slope of the surface in any direction can be taken equal to the angle 
that the tangent to the surface in that direction makes with the xy plane, 
and the square of the slope may be neglected compared to unity. The 
curvature of the surface in a plane parallel to the xz plane (Fig. 16) is 
then numerically equal to 

1 = _ A (?™\ d2w 

T z dx\dxj dx 2 

We consider a curvature positive if it is convex downward. The minus 
sign is taken in Eq. (e), since for the deflection convex downward, as 
shown in the figure, the second derivative d 2 w/dx 2 is negative. 

In the same manner we obtain for the curvature in a plane parallel to 
the yz plane 

1_ _ _ d_ /dw\ d 2 w 



(£) = - ^ w 



0- 



dy\dyj dy* KJJ 

These expressions are similar to those used in discussing the curvature 
of a bent beam. 

In considering the curvature of the middle surface in any direction an 
(Fig. 16) we obtain 

_1 = _ d_ /dw\ 

r n dn \dnj 

Substituting expression (a) for dw/dn and observing that 



we find 

1 



d d . d . 

— = -r- COS a + — sin a 
dn ox ay 



(B . d . \/dw . dw . \ 

l -r- cos a + — sm a J I -r- cos a + — sin a J 
\dx by ) \dx dy ) 

/d*w , . d*w . .d 2 w. 2 \ 

(w C ° S a + 2 dx~dy- Sm a C0S a + W Sma ) 



1 „ 1 . rt , 1 



cos 



a sin 2a -1 sin 2 a (g) 



r x r xy 



PURE PEXDIXG OF PLATES 35 

It is seen that the curvature in any direction n at a point of the middle 
surface can be calculated if we know at that point the curvatures 



and the quantity 



T x dx 2 r u djf 1 



dhv 


1 


dx- 


r u ' 


1 


dhv 



(h) 



r xu dx dy 

which is called the twist of the surface with respect to the x and y axes. 

If instead of the direction an (Fig. 10/;) we take the direction at per- 
pendicular to an, the curvature in this new direction will be obtained from 
expression (g) by substituting tt/2 + a for a. Thus we obtain 

— = — sin 2 a H sin 2a H cos 2 a (i) 

Tt r x r xi/ r y K J 

Adding expressions (g) and {%), we find 

I + i = I + I (34) 

which shows that at any point of the middle surface the sum of the 
curvatures in two perpendicular directions such as n and t is independent 
of the angle a. This sum is usually called the average curvature of the 
surface at a point. 

The twist of the surface at a with respect to the an and at directions is 



1_ = !L (<!™\ 
r nt dt \dnj 



In calculating the derivative with respect to t, we observe that the 
direction at is perpendicular to an. Thus we obtain the required deriva- 
tive by substituting tt/2 + a for a in Eq. (a). In this manner we find 



Tnt 



(d d . \ / dw . , dw \ 

— cos a + — sin a J ( — sin a + -7— cos a ) 
d.r dy J\ dx dy ) 

1 . / a 2 ™ aV\ . _ <9 2 ™ 

2 8m 2a V~ dF» + apj + cos 2a ^ 

- sin 2a ( J + cos 2a — 

2 V* TyJ r X y 



(J) 



In our further discussion we shall be interested in finding in terms of a 
the directions in which the curvature of the surface is a maximum or a 
minimum and in finding the corresponding values of the curvature. We 
obtain the necessary equation for determining a by equating the deriva- 
tive of expression (g) with respect to a to zero, which gives 

12 1 

— sin 2a H cos 2a sin 2a = (k) 

X I XV I u 



36 



THEORY OF PLATES AND SHELLS 



whence 



tan 2a = — 



r xy 






(35) 



From this equation we find two values of a, differing by ir/2. Substitut- 
ing these in Eq. (g) we find two values of l/r n , one representing the 
maximum and the other the minimum curvature at a point a of the sur- 
face. These two curvatures are called the principal curvatures of the 
surface; and the corresponding planes naz and taz, the principal planes of 
curvature. 

Observing that the left-hand side of Eq. (k) is equal to the doubled 
value of expression (j), we conclude that, if the directions an and at (Fig. 
16) are in the principal planes, the corresponding twist l/r nt is equal to 
zero. 

We can use a circle, similar to Mohr's circle representing combined 
stresses, to show how the curvature and the twist of a surface vary with 
the angle a.* To simplify the discussion we assume that the coordinate 
planes xz and yz are taken parallel to the principal planes of curvature 

at the point a. Then 

± = 

' XII 




•n 



and we obtain from Eqs. (g) and (j) 
for any angle a 



Fig. 17 



— = — cos 2 a H sin 2 a 

Tn Tx Ty 

1 1/1 1\ • 

— = n I ) sin 2a 

r„ t 2 \r x r y ) 



(36) 



Taking the curvatures as abscissas and the twists as ordinates and con- 
structing a circle on the diameter l/r« - l/r„, as shown in Fig. 17, we see 
that the point A defined by the angle 2a has the abscissa 



= — cos 2 a H sin 2 



a 



and the ordinate 



2 \r x r v ) 



AB = h « 1 sin 2a 



Comparing these results with formulas (36), we conclude that the coordi- 
* See S. Timoshenko, "Strength of Materials," part I, 3d ed., p. 40, 1955. 



PURE BENDING OF PLATES 



37 



nates of the point A define the curvature and the twist of the surface f or 
any value of the angle a. It is seen that the maximum twist, represented 
by the radius of the circle, takes place when a = tt/4, i.e., when we take 
two perpendicular directions bisecting the angles between the principal 
planes. 

In our example the curvature in any direction is positive; hence the 
surface is bent convex downward. If the curvatures \/r x and l/r y are 
both negative, the curvature in any direction is also negative, and we have 
a bending of the plate convex upward. Surfaces in which the curvatures 
in all planes have like signs are called synclastic. Sometimes we shall 
deal with surfaces in which the two principal curvatures have opposite 
signs. A saddle is a good example. Such surfaces are called anticlastic. 
The circle in Fig. 18 represents a particular case of sych surfaces when 







OB] 


1 






\l'7 


r n 




1 


-X 






—-1-* 


*.-! — 




•** 


r x 




' 







M v 



Fig. 18 




r Kf((((((((, h X 



r r r rr rr 




Mv 



Fig. 19 



\/r y = — l/r x . It is seen that in this case the curvature becomes zero 
for a = 7r/4 and for a = Sir/4:, and the twist becomes equal to + \/r x . 

10. Relations between Bending Moments and Curvature in Pure 
Bending of Plates. In the case of pure bending of prismatic bars a 
rigorous solution for stress distribution is obtained by assuming that 
cross sections of the bar remain plane during bending and rotate only 
with respect to their neutral axes so as to be always normal to the deflec- 
tion curve. Combination of such bending in two perpendicular directions 
brings us to pure bending of plates. Let us begin with pure bending of a 
rectangular plate by moments that are uniformly distributed along the 
edges of the plate, as shown in Fig. 19. We take the xy plane to coincide 
with the middle plane of the plate before deflection and the x and y axes 
along the edges of the plate as shown. The z axis, which is then per- 
pendicular to the middle plane, is taken positive downward. We denote 
by M x the bending moment per unit length acting on the edges parallel 
to the y axis and by M y the moment per unit length acting on the edges 
parallel to the x axis. These moments we consider positive when they 
are directed as shown in the figure, i.e., when they produce compression 



38 



THEOKY OF PLATES AND SHELLS 



dy 



^*— --dx 






in the upper surface of the plate and tension in the lower. The thickness 
of the plate we denote, as before, by h and consider it small in comparison 
with other dimensions. 

Let us consider an element cut out of the plate by two pairs of planes 
parallel to the xz and yz planes, as shown in Fig. 20. Since the case shown 
in Fig. 19 represents the combination of two uniform bendings, the stress 
conditions are identical in all elements, as shown in Fig. 20, and we have 

a uniform bending of the plate. Assuming 
that during bending of the plate the lateral 
sides of the element remain plane and rotate 
about the neutral axes nn so as to remain nor- 
mal to the deflected middle surface of the 
plate, it can be concluded that the middle 
plane of the plate does not undergo any ex- 
tension during this bending, and the middle 
surface is therefore the neutral surface. 1 Let 
\/r x and l/r y denote, as before, the curva- 
tures of this neutral surface in sections parallel to the xz and yz planes, 
respectively. Then the unit elongations in the x and y directions of an 
elemental lamina abed (Fig. 20), at a distance z from the neutral surface, 
are found, as in the case of a beam, and are equal to 

e y = - (a) 



"T*F 



v> 7ZZZZZZZ v ^j^H^ 



Ul_ 



dz 



h. 
2 

i 



h. 
Z 



Fig. 20 



z 
r x 



Using now Hooke's law [Eq. (1), page 5], the corresponding stresses in 
the lamina abed are 

Ez 



<*x = 



V* Ty) 

Ez ( 1 1\ 



(6) 



These stresses are proportional to the distance z of the lamina abed from 
the neutral surface and depend on the magnitude of the curvatures of the 
bent plate. 

The normal stresses distributed over the lateral sides of the element in 
Fig. 20 can be reduced to couples, the magnitudes of which per unit 
length evidently must be equal to the external moments M x and M y . In 
this way we obtain the equations 



J a x z dy dz = M' x dy 

r 



h/2 
h/2 

-h/2 



(c) 



dyZ dx dz 



My dx 



1 It will be shown in Art. 13 that this conclusion is accurate enough if the deflections 
of the plate are small in comparison with the thickness h. 



PURE BENDING OF PLATES 



Substituting expressions (6) for a x and <r„, we obtain 






d 2 w 

a* 2 

d 2 w 



+ 



+ 



d 2 W 
dy' 

d 2 w 
~dx 2 



39 



(37) 
(38) 



where D is the flexural rigidity of the plate defined by Eq. (3), and w 
denotes small deflections of the plate in the z direction. 

Let us now consider the stresses acting on a section of the lamina 
abed parallel to the z axis and inclined to the x and y axes. If acd (Fig. 21) 
represents a portion of the lamina cut by such a section, the stress acting 
on the side ac can be found by means of the equations of statics. Resolv- 
ing this stress into a normal component a n and a shearing component r n t, 





< — 


dx--. 


-> 




d 


iOy 




I 










^06 


^\ 




+ 




\ Tnt V n 





*^C 



^M nt 
n 



(a) 




Fig. 21 



the magnitudes of these components are obtained by projecting the forces 
acting on the element acd on the n and t directions respectively, which 
gives the known equations 



^n = cr x cos 2 a + (T y sin 2 a 
Tnt = $(jtr y — <j x ) sin 2a 



(<*) 



in which a is the angle between the normal n and the x axis or between 
the direction t and the y axis (Fig. 21a). The angle is considered positive 
if measured in a clockwise direction. 

Considering all laminas, such as acd in Fig. 216, over the thickness of 
the plate, the normal stresses <r n give the bending moment acting on the 
section ac of the plate, the magnitude of which per unit length along ac 
is 

M n = ]_ h/2 <?nz dz = M x cos 2 a + M y sin 2 a (39) 

The shearing stresses r nt give the twisting moment acting on the section 



40 THEORY OF PLATES AND SHELLS 

ac of the plate, the magnitude of which per unit length of ac is 

Mnt = - f h/ * r nt z dz=i sin 2a(M x - M v ) (40) 

J —h/2 

The signs of M n and M nt are chosen in such a manner that the positive 
values of these moments are represented by vectors in the positive direc- 
tions ol n and t (Fig. 21a) if the rule of the right-hand screw is used. 
When a is zero or tt, Eq. (39) gives M n = M x . For a = tt/2 or 3tt/2, we 

obtain M n = M y . The moments Af n < become 
zero for these values of a. Thus we obtain 
the conditions shown in Fig. 19. 

Equations (39) and (40) are similar to Eqs. 
-Mn (36), and by using them the bending and 
twisting moments can be readily calculated 
^_^ for any value of a. We can also use the 

nt graphical method for the same purpose and 

Fia 22 find the values of M n and M nt from Mohr's 

circle, which can be constructed as shown in the previous article by tak- 
ing M n as abscissa and M nt as ordinate. The diameter of the circle will 
be equal to M x — M VJ as shown in Fig. 22. Then the coordinates OB and 
~AB of a point A, defined by the angle 2a, give the moments M n and M nt 
respectively. 

Let us now represent M n and M nt as functions of the curvatures and 
twist of the middle surface of the plate. Substituting in Eq. (39) for 
M x and M y their expressions (37) and (38), we find 

M n - D (- cos 2 a + - sin 2 a) + vD ( - sin 2 a + - cos 2 a) 

\r x Ty / \T X Ty / 

Using the first of the equations (36) of the previous article, we conclude 
that the expressions in parentheses represent the curvatures of the middle 
surface in the n and t directions respectively. Hence 

To obtain the corresponding expression for the twisting moment M nt , 
let us consider the distortion of a thin lamina abed with the sides ab and 
ad parallel to the n and t directions and at a distance z from the middle 
plane (Fig. 23). During bending of the plate the points a, 6, c, and d 
undergo small displacements. The components of the displacement of 
the point a in the n and t directions we denote by u and v respectively. 
Then the displacement of the adjacent point d in the n direction is 
u + (du/dt) dt, and the displacement of the point b in the t direction is 
v + (dv/dn) dn. Owing to these displacements, we obtain for the shear- 



PURE BENDING OF PLATES 



41 



ing strain 



_ du dv 



The corresponding shearing stress is 

- fl (£ + £) 



T n t 



W 



(/) 



From Fig. 236, representing the section of the middle surface made by 
the normal plane through the n axis, it may be seen that the angle of 
rotation in the counterclockwise direction of an element pq, which 
initially was perpendicular to the xy plane, about an axis perpendicular 
to the nz plane is equal to — dw/dn. Owing to this rotation a point of the 



V^A. 




Fig. 23 

element at a distance z from the neutral surface has a displacement in the 
n direction equal to 

dw 



u = 



dn 



Considering the normal section through the t axis, it can be shown that 
the same point has a displacement in the t direction equal to 



v = — z 



dw 
~dt 



Substituting these values of the displacements u and v in expression (/), 
we find 

d 2 w 



T n t = ~2GZ 



dn dt 



and expression (40) for the twisting moment becomes 



M. t 



nt 



i- 



h ' 2 , Gh 3 d*w 

T nt Z dz = -=- ^—^ = D(l 
h/2 § dn dt 



d 2 w 
'Hindi 



(42) 



(43) 



42 THEORY OF JLATES AND SHELLS 

It is seen that the twisting moment for the given perpendicular directions 
n and t is proportional to the twist of the middle surface corresponding to 
those directions. When the n and t directions coincide with the x and 
y axes, there are only bending moments M x and M y acting on the sections 
perpendicular to those axes (Fig. 19). Hence the corresponding twist is 
zero, and the curvatures \/r x and \/r y are the principal curvatures of the 
middle surface of the plate. They can readily be calculated from 
Eqs. (37) and (38) if the bending moments M x and M y are given. The 
curvature in any other direction, defined by an angle a, can then be 
calculated by using the first of the equations (36), or it can be taken from 
Fig. 17. 

Regarding the stresses in a plate undergoing pure bending, it can be 
concluded from the first of the equations (d) that the maximum normal 
stress acts on those sections parallel to the xz or yz planes. The magni- 
tudes of these stresses are obtained from Eqs. (6) by substituting z = h/2 
and by using Eqs. (37) and (38). In this way we find 

- ®M± ( \ - ®Ml (AA\ 

(/7"x)max 7 2 \P"u) max 70 V"/ 

If these stresses are of opposite sign, the maximum shearing stress acts in 
the plane bisecting the angle between the xz and yz planes and is equal to 

If the stresses (44) are of the same sign, the maximum shear acts in the 
plane bisecting the angle between the xy and xz planes or in that bisecting 
the angle between the xy and yz planes and is equal to %(<T y )mnx or ^(cr^max, 
depending on which of the two principal stresses (o^ma* or (a z ) m ^ is 
greater. 

11. Particular Cases of Pure Bending. In the discussion of the previ- 
ous article we started with the case of a rectangular plate with uniformly 
distributed bending moments acting along the edges. To obtain a gen- 
eral case of pure bending of a plate, let us imagine that a portion of any 
shape is cut out from the plate considered above (Fig. 19) by a cylindrical 
or prismatic surface perpendicular to the plate. The conditions of bend- 
ing of this portion will remain unchanged provided that bending and 
twisting moments that satisfy Eqs. (39) and (40) are distributed along the 
boundary of the isolated portion of the plate. Thus we arrive at the 
case of pure bending of a plate of any shape, and we conclude that pure 
bending is always produced if along the edges of the plate bending 
moments M n and twisting moments M nt are distributed in the manner 
given by Eqs. (39) and (40). 

Let us take, as a first example, the particular case in which 

M x = M y = M 



PURE BENDING OF PLATES 43 

It can be concluded, from Eqs. (39) and (40), that in this case, for a plate 
of any shape, the bending moments are uniformly distributed along the 
entire boundary and the twisting moments vanish. From Eqs. (37) and 
(38) we conclude that 

i = i = M un 

r x r, D{\ + v) [ ° j 

i.e., the plate in this case is bent to a spherical surface the curvature of 
which is given by Eq. (40). 

In the general case, when M x is different from M y , we put 

M x = M\ and M y = M 2 

Then, from Eqs. (37) and (38), we find 

d 2 w _ Mr- vM* 

ax* ~ D(T - ^ 

d 2 w = _ Mjj- vMi K 

~dy 2 " Z)(f- V 2 ) 

and in addition 

ox oy 
Integrating these equations, we find 

Mi — vM<i 9 Mi — vM i 9 ~ ~ ~ / v 

w = - 2DTr=^y *" ~ 2D(r-75) r + c >* + ° 22/ + Cs (c) 

where d, C 2 , and C 3 are constants of integration. These constants 
define the plane from which the deflections w are measured. If this 
plane is taken tangent to the middle surface of the plate at the origin, 
the constants of integration must be equal to zero, and the deflection 
surface is given by the equation 

In the particular case where Mi = M 2 = M, we get from Eq. (d) 

i.e., a paraboloid of revolution instead of the spherical surface given by 
Eq. (46). The inconsistency of these results arises merely from the use 
of the approximate expressions d 2 w/dx 2 and d 2 w/dy 2 for the curvatures 
l/r z and \/r y in deriving Eq. (e). These second derivatives of the 
deflections, rather than the exact expressions for the curvatures, will be 
used also in all fuither considerations, in accordance with the assump- 
tions made in Art. 9. This procedure greatly simplifies the fundamental 
equations of the theory of plates. 



44 THEORY OF PLATES AND SHELLS 

Returning now to Eq. (d), let us put M 2 = — Mi. In this ease the 
principal curvatures, from Eqs. (a), are 

1 1 d*w Mi 



r x r v dx* D(l - v) 

and we obtain an anticlastic surface the equation of which is 



(/) 




Straight lines parallel to the x axis become, after bending, parabolic 
curves convex downward (Fig. 24), whereas straight lines in the y direc- 
tion become parabolas convex upward. Along the lines bisecting the 
angles between the x and y axes we have x = y, or x = —y; thus deflec- 
tions along these lines, as seen from Eq. (g), are zero. All lines parallel 
to these bisecting lines before bending remain straight during bending, 
rotating only by some angle. A rectangle abed bounded by such lines 

will be twisted as shown in Fig. 24. 
Imagine normal sections of the plate 
along lines ab, be, ed, and ad. From 
Eqs. (39) and (40) we conclude that 
bending moments along these sections 
are zero and that twisting moments 
along sections ad and be are equal to 
24 Mi and along sections ab and ed are 

equal to — Mi. Thus the portion abed of the plate is in the condition of 
a plate undergoing pure bending produced by twisting moments uni- 
formly distributed along the edges (Fig. 25a). These twisting moments 
are formed by the horizontal shearing stresses continuously distributed 
over the edge [Eq. (40)]. This horizontal stress distribution can be 
replaced by vertical shearing forces which produce the same effect as 
the actual distribution of stresses. To show this, let the edge ab be 
divided into infinitely narrow rectangles, such as mnpq in Fig. 256. If 
A is the small width of the rectangle, the corresponding twisting couple 
is MiA and can be formed by two vertical forces equal to M x acting along 
the vertical sides of the rectangle. This replacement of the distributed 
horizontal forces by a statically equivalent system of two vertical forces 
cannot cause any sensible disturbance in the plate, except within a distance 
comparable with the thickness of the plate, 1 which is assumed small. 
Proceeding in the same manner with all the rectangles, we find that all 
forces Mi acting along the vertical sides of the rectangles balance one 
another and only two forces Mi at the corners a and d are left. Making 

1 This follows from Saint VenanVs principle; see S. Timoshenko and J. N. Goodier, 
'Theory of Elasticity," 2d ed., p. 33, 1951. 



PURE BENDING OF PLATES 



45 



the same transformation along the other edges of the plate, we conclude 
that bending of the plate to the anticlastic surface shown in Fig. 25a can 
be produced by forces concentrated at the corners 1 (Fig. 25c). Such an 
experiment is comparatively simple to perform, and was used for the 
experimental verification of the theory of bending of plates discussed 
above. 2 In these experiments the deflections of the plate along the line bod 
(Fig. 24) were measured and were found to be in very satisfactory agree- 
ment with the theoretical results obtained from Eq. (g). Some dis- 
crepancies were found only near the edges, and they were more pro- 





Fig. 25 

nounced in the case of comparatively thick plates, as would be expected 
from the foregoing discussion of the transformation of twisting couples 
along the edges. 

As a last example let us consider the bending of a plate (Fig. 19) to a 
cylindrical surface having its generating line parallel to the y axis, 
such a case d 2 w/dy 2 = 0, and we find, from Eqs. (37) and (38), 



±11 



M x = -D 



d 2 w 
~dx 2 



M„ 



-pD 



d 2 w 
dx 2 



(h) 



It is seen that to produce bending of the plate to a cylindrical surface 
we must apply not only the moments M x but also the moments M y . 
Without these latter moments the plate will be bent to an anticlastic 
surface. 3 The first of equations (h) has already been used in Chap. 1 in 
discussing the bending of long rectangular plates to a cylindrical surface. 
Although in that discussion we had a bending of plates by lateral loads 
and there were not only bending stresses but also vertical shearing stresses 

1 This transformation of the force system acting along the edges was first suggested 
by Lord Kelvin and P. G. Tait; see "Treatise on Natural Philosophy/' vol. 1, part 2, 
p. 203, 1883. 

2 Such experiments were made by A. Nddai, Forschungsarb., vols. 170, 171, Berlin, 
1915; see also his book "Elastische Platten," p. 42, Berlin, 1925. 

3 We always assume very small deflections or else bending to a developable surface. 
The case of bending to a nondevelopable surface when the deflections are not small 
will be discussed later; see p. 47. 



46 THEORY OF PLATES AND SHELLS 

acting on sections perpendicular to the x axis, it can be concluded from a 
comparison with the usual beam theory that the effect of the shearing 
forces is negligible in the case of thin plates, and the equations developed 
for the case of pure bending can be used with sufficient accuracy for 
lateral loading. 

12. Strain Energy in Pure Bending of Plates. If a plate is bent by 
uniformly distributed bending moments M x and M y (Fig. 19) so that the 
xz and yz planes are the principal planes of the deflection surface of the 
plate, the strain energy stored in an element, such as shown in Fig. 20, 
is obtained by calculating the work done by the moments M x dy and M y dx 
on the element during bending of the plate. Since the sides of the ele- 
ment remain plane, the work done by the moments M x dy is obtained by 
taking half the product of the moment and the angle between the corre- 
sponding sides of the element after bending. Since —d 2 w/dx z represents 
the curvature of the plate in the xz plane, the angle corresponding to the 
moments M x dy is — (d 2 w/dx 2 ) dx, and the work done by these moments is 



1 . . d 2 w 

2 M *M 



— o M * -^~2 dx dy 



An analogous expression is also obtained for the work produced by the 

moments M y dx. Then the total work, equal to the strain energy of the 

element, is 

1 /., d 2 w . „, d 2 w\ , , 
dV =-2(M x — 2 + M v — 2 )dxdy 

Substituting for the moments their expressions (37) and (38), the strain 
energy of the elements is represented in the following form : 






dx dy (a) 



Since in the case of pure bending the curvature is constant over the 
entire surface of the plate, the total strain energy of the plate will be 
obtained if we substitute the area A of the plate for the elementary area 
dx dy in expression (a). Then 

r-wKSO"'©^^] (47) 

If the directions x and y do not coincide with the principal planes of 
curvature, there will act on the sides of the element (Fig. 20) not only 
the bending moments M x dy and M y dx but also the twisting moments 
M xy dy and M yx dx. The strain energy due to bending moments is repre- 
sented by expression (a). In deriving the expression for the strain energy 
due to twisting moments M xy dy we observe that the corresponding angle 
of twist is equal to the rate of change of the slope dw/dy, as x varies, 



PURE BENDING OF PLATES 47 

multiplied with dx; hence the strain energy due to M xy dy is 

1 d*W 

2 Mxv dx^y dx dy 



which, applying Eq. (43), becomes 

I 
2 



D( ' -" ) fe) !rfc *' 



The same amount of energy will also be produced by the couples M yx dx, 
so that the strain energy due to both twisting couples is 

D(l - p) (^fX dx dy (b) 



Since the twist does not affect the work produced by the bending 
moments, the total strain energy of an element of the plate is obtained by 
adding together the energy of bending (a) and the energy of twist (6). 
Thus we obtain 

dV = 2 D l(w) + {!?) + 2V ^ W\ dX dV 
or 

(48) 

The strain energy of the entire plate is now obtained by substituting 
the area A of the plate for the elemental area dx dy. Expression (48) 
will be used later in more complicated cases of bending of plates. 

13. Limitations on the Application of the Derived Formulas. In dis- 
cussing stress distribution in the case of pure bending (Art. 10) it was 
assumed that the middle surface is the neutral surface of the plate. This 
condition can be rigorously satisfied only if the middle surface of the bent 
plate is a developable surface. Considering, for instance, pure bending of 
a plate to a cylindrical surface, the only limitation on the application of 
the theory will be the requirement that the thickness of the plate be 
small in comparison with the radius of curvature. In the problems of 
bending of plates to a cylindrical surface by lateral loading, discussed in 
the previous chapter, it is required that deflections be small in compari- 
son with the width of the plate, since only under this condition will the 
approximate expression used for the curvature be accurate enough. 

If a plate is bent to a nondevelopable surface, the middle surface 
undergoes some stretching during bending, and the theory of pure bend- 



48 



THEORY OF PLATES AND SHELLS 



ing developed previously will be accurate enough only if the str 
corresponding to this stretching of the middle surface are small in 
parison with the maximum bending stresses given by Eqs. (44) or, 
is equivalent, if the strain in the middle surface is small in compa 
with the maximum bending strain h/2r milx . This requirement pu1 
additional limitation on deflections of a plate, viz., that the deflectic 
of the plate must be small in comparison with its thickness h. 

To show this, let us consider the bending of a circular plate by t 
ing couples M uniformly distributed along the edge. The deflection 
face, for small deflections, is spherical with radius r as defined by Eq. 
Let AOB (Fig. 26) represent a diametral section of the bent circular j 
a its outer radius before bending, and 6 the deflection at the middle. 
assume at first that there is no stretching of the middle surface o 
plate in the radial direction. In such a case the arc OB must be equ 
the initial outer radius a of the plate. The angle <p and the radius 
the plate after bending are then given by the following equations: 



a 

<p = - 
r 



b = r sin <p 



It is seen that the assumed bending of the plate implies a compre 
strain of the middle surface in the circumferential direction. The ms 

tude of this strain at the edge of the plate i 



M 









a — b r<p — r sin cp 






a r<p 


-*- 


V 


For small deflections we can take 


6 


c \ 



M 


sin (p = (p — y 
which, substituted in Eq. (a), gives 


Fig. 


26 




<P 2 
e = -6" 



To represent this strain as a function of the maximum deflection I 
observe that 



b = r(l — cos ip) 
25 



Tip 1 



Hence <p 

Substituting in Eq. (6), we obtain 



2 — 



€ = 



3r 



PURE BENDING OF PLATES 4.9 

This represents an upper limit for the circumferential strain at the edge 
of the plate. It was obtained by assuming that the radial strain is zero. 
Under actual conditions there is some radial strain, and the circumfer- 
ential compression is somewhat smaller 1 than that given by Eq. (49). 

From this discussion it follows that the equations obtained in Art. 10, 
on the assumption that the middle surface of the bent plate is its neutral 
surface, are accurate provided the strain given by expression (49) is small 
in comparison with the maximum bending strain h/2r, or, what is equiva- 
lent, if the deflection S is small in comparison with the thickness h of the 
plate. A similar conclusion can also be obtained in the more general 
case of pure bending of a plate when the two principal curvatures are 
not equal. 2 Generalizing these conclusions we can state that the equa- 
tions of Art. 10 can always be applied with sufficient accuracy if the 
deflections of a plate from its initial plane or from a true developable 
surface are small in comparison with the thickness of the plate. 

14. Thermal Stresses in Plates with Clamped Edges. Equation (46) 
for the bending of a plate to a spherical surface can be used in calculating 
thermal stresses in a plate for certain cases of nonuniform heating. 
Assume that the variation of the temperature through the thickness of 
the plate follows a linear law and that the temperature does not vary in 
planes parallel to the surfaces of the plate. In such a case, by measuring 
the temperature with respect to that of the middle surface, it can be 
concluded that temperature expansions and contractions are proportional 
to the distance from the middle surface. Thus we have exactly the same 
condition as in the pure bending of a plate to a spherical surface. If the 
edges of the nonuniformly heated plate are entirely free, the plate will 
bend to a spherical surface. 3 Let a be the coefficient of linear expansion 
of the material of the plate, and let t denote the difference in temperature 
of the upper and lower faces of the plate. The difference between the 
maximum thermal expansion and the expansion at the middle surface is 
at/2, and the curvature resulting from the nonuniform heating can be 
found from the equation 

at h z N 

J-Tr (a) 

from which 

7 - J ™ 

This bending of the plate does not produce any stresses, provided the 

1 This question is discussed later; see Art. 96. 

2 See Kelvin and Tait, op. cit., vol. 1, part 2, p. 172. 

3 It is assumed that deflections are small in comparison with the thickness of the 
plate. 



50 THEORY OF PLATES AND SHELLS 

edges are free and deflections are small in comparison with the thickness 
of the plate. 

Assume now that the middle plane of the plate is free to expand but 
that the edges are clamped so that they cannot rotate. In such a case 
the nonuniform heating will produce bending moments uniformly dis- 
tributed along the edges of the plate. The magnitude of these moments 
is such as to eliminate the curvature produced by the nonuniform heat- 
ing [Eq. (50)], since only in this way can the condition at the clamped 
edge be satisfied. Using Eq. (46) for the curvature produced by the 
bending moments, we find for determining the magnitude M of the 
moment per unit length of the boundary the equation 1 

M at 



Z)(l + v) h 
from which 

m = <ML±A (b) 

The corresponding maximum stress can be found from Eqs. (44) and is 
equal to 

6M QatDjl + v) 

Substituting for D its expression (3), we finally obtain 

atE 



2(1 - v) 



(51) 



It is seen that the stress is proportional to the coefficient of thermal 
expansion a, to the temperature difference t between the two faces of 
the plate, and to the modulus of elasticity E. The thickness h of the 
plate does not enter into formula (51); but since the difference t of tem- 
peratures usually increases in proportion to the thickness of the plate, it 
can be concluded that greater thermal stresses are to be expected in thick 
plates than in thin ones. 

1 The effect of pure bending upon the curvature of the entire plate thus is equivalent 
but opposite in sign to the effect of the temperature gradient. Now, if the plate 
remains, in the end, perfectly plane, the conditions of a built-in edge are evidently 
satisfied along any given boundary. Also, since in our case the bending moments are 
equal everywhere and in any direction, the clamping moments along that given 
boundary are always expressed by the same Eq. (6). 



CHAPTER 3 



SYMMETRICAL BENDING OF CIRCULAR PLATES 



15. Differential Equation for Symmetrical Bending of Laterally Loaded 
Circular Plates. 1 If the load acting on a circular plate is symmetrically 
distributed about the axis perpendicular to the plate through its center, 
the deflection surface to which the middle plane of the plate is bent will 
also be symmetrical. In all points equally distant from the center of 
the plate the deflections will be the same, and it is sufficient to consider 
deflections in one diametral section through the axis of symmetry (Fig. 
27). Let us take the origin of coordinates 
at the center of the undeflected plate and 
denote by r the radial distances of points 
in the middle plane of the plate and by w 
their deflections in the downward direction. 
The maximum slope of the deflection sur- 
face at any point A is then equal to — dw/dr, 
and the curvature of the middle surface of 
the plate in the diametral section rz for 
small deflections is 



d 2 w 



d<p 
dr 



(a) 




where <p is the small angle between the normal to the deflection surface 
at A and the axis of symmetry OB. From symmetry we conclude that 
l/r n is one of the principal curvatures of the deflection surface at A. 
The second principal curvature will be in the section through the normal 
AB and perpendicular to the rz plane. Observing that the normals, such 
as AB, for all points of the middle surface with radial distance r form a 
conical surface with apex B, we conclude that the length AB is the radius 
of the second principal curvature which we denote by r*. Then, from 
the figure, we obtain 

1 1 dw (p ,- N 



r t 



r dr 



r 



1 The solution of these problems of bending of circular plates was given by Poisson; 
see "Memoirs of the Academy/' vol. 8, Paris, 1829. 

51 



52 



THEORY OF PLATES AND SHELLS 



Having expressions (a) and (b) for the principal curvatures, we can obtain 
the corresponding values of the bending moments assuming that relations 
(37) and (38), derived for pure bending, also hold between these moments 
and the curvatures. 1 Using these relations, we obtain 



M r = 

M t = 



-d( 



r dr J \dr r ) 



d 2 w 



\dr 2 



(52) 
(53) 



where, as before, M r and M t denote bending moments per unit length. 

The moment M r acts along circumferential sections of the plate, such as 

the section made by the conical surface with the apex at B, and M t acts 

along the diametral section rz of the plate. 

Equations (52) and (53) contain only one variable, w or <p, which can 

be determined by considering the equilibrium of an element of the plate 

such as element abed in Fig. 28 cut out 
from the plate by two cylindrical sec- 
tions ab and cd and by two diametral 
sections ad and be. The couple acting 
on the side cd of the element is 






^r> 




M r rde 



(<0 



Fig. 28 



The corresponding couple on the side 
ab is 

(M r + ^ dr\ (r + dr) dO (d) 



The couples on the sides ad and be of the element are each M t dr, and they 
give a resultant couple in the plane rOz equal to 



M t dr dO 



(e) 



From symmetry it can be concluded that the shearing forces that may 
act on the element must vanish on diametral sections of the plate but 
that they are usually present on cylindrical sections such as sides ed and 
ab of the element. Denoting by Q the shearing force per unit length of 

1 The effect on deflections of shearing stresses acting on normal sections of the plate 
perpendicular to meridians, such as the section cut by the conical surface with the 
apex at B, is neglected here. Their effect is slight in the case of plates in which the 
thickness is small in comparison with the diameter. Further discussion of this subject 
will be given in Art. 20. The stresses perpendicular to the surface of the plate are 
also neglected, which is justifiable in all cases when the load is not highly concentrated 
(see p. 69). 



SYMMETRICAL BENDING OF CIRCULAR PLATES 



53 



the cylindrical section of radius r, the total shearing force acting on the 
side cd of the element is Qr dO, and the corresponding force on the side ab is 



Q + 



©*] 



(r + dr) dd 



Neglecting the small difference between the shearing forces on the two 
opposite sides of the element, we can state that these forces give a couple 
in the rz plane equal to 

Qr dd dr (/) 

Summing up the moments (c), (d), (e), and (/) with proper signs and 
neglecting the moment due to the external load on the element as a 
small quantity of higher order, we obtain the following equation of 
equilibrium of the element abed: 

( Mr + ^l dr \ ( r + dr) d6 - M r r dd - M t dr dd + Qr dd dr = 

from which we find, by neglecting a small quantity of higher order, 

Mr + ^r-M t + Qr = (g) 

dr 

Substituting expressions (52) and (53) for M r and M t , Eq. (g) becomes 



or, in another form, 



d 2 <p 1 dip 
dr 2 r dr 



d*w 1 d 2 w 



<P 



dr z 



r dr 2 



1 dw 
r 2 dr 



9. 

D 

Q 
D 



(54) 



(55) 



In any particular case of a symmetrically loaded circular plate the 
shearing force Q can easily be calculated by dividing the load distributed 
within the circle of radius r by 2irr; then Eq. (54) or (55) can be used to 
determine the slope <p and the deflection w of the plate. The integration 
of these equations is simplified if we observe that they can be put in the 
following forms: 



d_ 
dr 

d_ 
dr 



-ri M } 

ld_ 

r dr 



9 
D 



X dr)\ D 



(56) 
(57) 



If Q is represented by a function of r, these equations can be integrated 
without any difficulty in each particular case. 

Sometimes it is advantageous to represent the right-hand side of Eq. 
(57) as a function of the intensity q of the load distributed over the plate. 
For this purpose we multiply both sides of the equation by 2irr. Then, 



54 THEORY OF PLATES AND SHELLS 

observing that 

Q2irr = r q2wr dr 



we obtain 



d 
dr 



[&('%]- tt** 



Differentiating both sides of this^equation with respect to r and dividing 
by r, we finally obtain 

rdr\ r drirdr\ dr J J J D 

This equation can easily be integrated if the intensity of the load q is 
given as a function of r. 

16. Uniformly Loaded Circular Plates. If a circular plate of radius a 
carries a load of intensity q uniformly distributed over the entire surface 
of the plate, the magnitude of the shearing force Q at a distance r from 
the center of the plate is determined from the equation 

2irrQ = irr 2 q 
from which 

o = f w 

Substituting in Eq. (57), we obtain 

L r dr \ dr ) J 
By one integration we find 



&r\~rdr\ r dr ) \ 2D 



(b) 



U/ dw\ 
r dr\ dr J 



A 



S + * « 



where d is a constant of integration to be found later from the conditions 
at the center and at the edge of the plate. Multiplying both sides of 
Eq. (c) by r, and making the second integration, we find 







dw _ 
dr 


qr* 
WD 


+ 


dr 2 
2 


■ + C- 


and 




dw _ 
dr 


qr 3 
16Z> 


+ 


C,r 
2 


r 


The 


new 


integration then gives 









(59) 



Let us now calculate the constants of integration for various particular 
cases. 



SYMMETRICAL BENDING OF CIRCULAR PLATES 55 

Circular Plate with Clamped Edges. In this case the slope of the deflec- 
tion surface in the radial direction must be zero f or r = and r = a. 
Hence, from Eq. (59), 



/ <p* C,r ck _ 



From the first of these equations we conclude that C 2 = 0. Substituting 
this in the second equation, we obtain 

With these values of the constants, Eq. (59) gives the following expres- 
sion for the slope: 

• --£-!&<•"-"> < 61 > 

Equation (60) gives 

or^ oa^r^ 

At the edge of the plate the deflection is zero. Hence, 

?« 4 _ 9* + c 3 = o 



64D 32D 
and we obtain 



c, = «° 4 



64D 

Substituting in Eq. (d), we find 

W = 64D (a2 " r2)2 (62) 

The maximum deflection is at the center of the plate and, from Eq. (62), 
is equal to 

w - 3°L W 

max ~~ 64D 

This deflection is equal to three-eighths of the deflection of a uniformly 
loaded strip with built-in ends having a flexural rigidity equal to D, a 
width of unity, and a length equal to the diameter of the plate. 

Having expression (61) for the slope, we obtain now the bending 
moments M r and M t by using expressions (52) and (53), from which 
we find 

Mr= lQ [a2(1 + v) ~ r2(3 + V)] (63) 

M t = ^t [a 2 (l + v) - r 2 (l + 3»)] (64) 



56 



THEORY OF PLATES AND SHELLS 



Substituting r = a in these expressions, we find for the bending moments 
at the boundary of the plate 



(M r )r-a = ~ 



qa 1 



(Mt)r-a = - 



vqa' 
~8~ 



At the center of the plate where r = 0, 



M r = M t 



&(!+» 



(65) 



(66) 



From expressions (65) and (66) it is seen that the maximum stress is at 
the boundary of the plate where 



\0V/max 



6M r 3 qa 2 



h 2 



4 h 2 



(/) 



The variation of stresses <r r and <r t at the lower face of the plate along the 
radius of the plate is shown in Fig. 29. 



o>; 


°t 


1 1 


Np.628^ 


s^0.827 

rs i 




i 

3(l+u')qo 2 


1 






8h 2 

1 
1 


/4 U h2 





1 


i 1 

r 
a 


1 1 1 X 1 

0.5 \ 
— ► ^ 


i ^^^ i 


i 


— i. 1 

1 

3 q'o 2 
"4T 2 " 



Fig. 29 

Circular Plate with Supported Edges. In calculating deflections for this 
case we apply the method of superposition. It was shown that in the 
case of clamped edges there are negative bending moments M r = — qa 2 /S 
acting along the edge (Fig. 30a). If this case is combined with that of 
pure bending shown in Fig. 306, the bending moments M r at the edge 
will be eliminated, and we obtain the bending of a plate supported at the 
edge. The deflection surface in the case of pure bending by the moments 
oa 2 /8, from Eq. (46) or Eq. (e) on page 43, is 



w = 



qa 2 



(a 2 - r 2 ) 



16D(1 + v) 

Adding this to the deflections (62) of the clamped plate, we find for the 
plate with a simply supported edge 



SYMMETRICAL BENDING OF CIRCULAR PLATES 

q(a 2 — r 2 ) 



w = 



64Z) 



(m---) 



57 

(67) 



Substituting r = in this expression we obtain the deflection of the plate 
at the center: 

= (5 + v)qa* 
w ™* 64(1 + v)D 

For v = 0.3 this deflection is about four times as great as that for the 
plate with clamped edge. 



(68) 




In calculating bending moments in this case we must add the constant 
bending moment ga 2 /8 to the moments (63) and (64) found above for the 
case of clamped edges. Hence in the case of supported edges 



Mr = j^(3+ ^)(a 2 -r 2 ) 

M t = ^ [a 2 (3 + v) - r 2 (l + 3*0] 

The maximum bending moment is at the center of the plate where 

M r = M t = ^±-" <?a 2 
16 

The corresponding maximum stress is 

<SM r _ 3(3 + v)qa* 



(69) 
(70) 



(O'r)max — (<^()m»x — 



h°- 



8/i 2 



(71) 



To get the maximum stress at any distance r from the center we must 
add to the stress calculated for the plate with clamped edges the con- 
stant value 

6 qa 2 

F 2 T 



58 



THEORY OF PLATES AND SHELLS 



corresponding to the pure bending shown in Fig. 306. The same stress is 
obtained also from Fig. 29 by measuring the ordinates from the horizontal 
axis through Oi. It may be seen that by clamping the edge a more 
favorable stress distribution in the plate is obtained. 

17. Circular Plate with a Circular Hole at the Center. Let us begin 
with a discussion of the bending of a plate by the moments Mi and M 2 






M 2 



Fig. 31 

uniformly distributed along the inner and outer boundaries, respectively 
(Fig. 31). The shearing force Q vanishes in such a case, and Eq. (57) 
becomes 

±\l±(r*°\\=f\ 
dr [ r dr\ dr ) J 

By integrating this equation twice we obtain 

dw C\r C 2 

Integrating again, we find the deflection 



(a) 



2 



*x 



10 = - ^ - C 2 log £ + C z 



(&) 



The constants of integration are now to be determined from the condi- 
tions at the edges. Substituting expression (a>intoOEq. (52), we find 



,, wfft C*]_ {Ci CA 



(c) 



This moment must be equal Ob-Mi for r = Vara' equal to M 2 for r = a. 
Hence equations for determining constants C\ and C 2 are 



from which 






= Mi 
= Mi 



Ci = 



2(a 2 M 2 - 6 2 Mi) 



C 2 = 



aWiMi - Mi) 



(d) 



(1 + v)D(a 2 - V) ~* (1 - v)D(a* - V) 
To determine the constant C 3 in Eq. (6), the deflections at the edges 



SYMMETRICAL BENDING OF CIRCULAR PLATES 59 

of the plate must be considered. Assume, for example, that the plate in 
Fig. 31 is supported along the outer edge. Then w = f 01 r = a, and 
we find, from (6), 

Cia? a 2 (a 2 M 2 - &WQ 
° 3 ~ 4 2(1 + v)D(a* - b 2 ) 

In the particular case when Mi = Owe obtain 

n _ 2b 2 M 1 n _ aWM x 

W — — T^ — i — _.\ ri/_9 Oa 02 — — 



(1 + v)D(a* - b 2 ) ' (1 - v)D(a 2 - b 2 ) 

a*b*Mi. 

Cs 2(1 + v)D{a* - b 2 ) 

and expressions (a) and (b) for the slope and the deflection become 

dw aWMx (\ 1 - v r\ { . , 

dr D(l - »)(a 2 - b 2 ) \r "^ 1 + v a 2 ) K } 

b i M 1 , 2 2 , , a 2 6 2 M! . r ,__. 

W = ~ 2(1 + v)D(a> - b 2 ) (a ~ r) + (1 - ,)D(a 2 - b 2 ) log a (73) 

As a second example we consider the case of bending of a plate by 
shearing forces Q uniformly distributed along the inner edge (Fig. 32). 
The shearing force per unit length of a 

circumference of radius r is r a 4 

| 4 b H- | 

Q ^ 0^ = _P_ ^zzzzzzzzzm j VZZZZZZZZZZZ2^ 

where P = 27r&Q denotes the total load^^ F g 32 

applied to the inner boundary of the 

plate. Substituting this in Eq. (57) and integrating, we obtain 

dw Pr / 01 r A Cir C 2 
^ = 8^V 2l0g a-V-"T~T 

mid ™ = S ( log s ~ " ^r " C2 log I + Cs 

The constants of integration will now be calculated from the boundary 
conditions. Assuming that the plate is simply supported along the outer 
edge, we have 

For the inner edge of the plate we have 

^ /d 2 w . v dw\ . ,, N 



(e) 

(/) 



60 THEORY OF PLATES AND SHELLS 

Substituting expressions (e) and (/) in Eqs. (g) and (h), we find 

2b 2 



C 2 = 
C 3 = 



4*D \1 



— v 



+ v 
(1 + v)P 



a 2 -b 
a 2 b 2 



2 log 



9 



(1 
8xD 



( 



v)^D a 2 



2 1 + 



F> log a 



a 2 — V 



(0 



log 



9 



With these values of the constants substituted in expressions (e) and (/), 
we find the slope and the deflection at any point of the plate shown in 
Fig. 32. For the slope at the inner edge, which will be needed in the 
further discussion, we obtain 



(*>\ =-™r 2 io g ^ 

\dr) r=b 8»DL a 



1 - 



1 - v 

1 + v 



, 2b 2 . b (. , a 2 1 + A ] ( - 



In the limiting case where b is infinitely small, b 2 log (b/a) approaches 
zero, and the constants of integration become 



d = 



1 - v P 



C 2 = 



8tD \ ^ 2 1 + v) 



T^~v±trD -» - " ~ a SttDV ' 21 + 
Substituting these values in expression (/), we obtain 

3 + v 



w 



8ttD L 



___^ (a2 _ r2)+r21 og- 



(*) 



This coincides with the deflection of a plate without a hole and loaded at 

the center [see Eq. (89), page 68]. Thus a very small hole at the center 

does not affect the deflection of the plate. 

Combining the loadings shown in Figs. 31 
and 32, we can obtain the solution for the 
case of a plate built in along the inner edge 
and uniformly loaded along the outer edge 
(Fig. 33). Since the slope at the built-in 
edge is zero in this case, using expressions 

(72) and (J), we obtain the following equation for determining the bending 

moment Mi at the built-in edge: 




D(l 



aWMx ( 1 1 - v b\ = Pb_ \ 

- v){a? - b 2 ) \b + 1 + v a 2 ) SwD [ 



2 log - - 1 



a 



1 + v 



+ 



2b 2 . b (. , a 2 l + v\ \ 
^=T> l0g a V 1 + b 2 1=1) \ 



SYMMETRICAL BENDING OF CIRCULAR PLATES 



61 



from which 
Mi = - 



47T 



(1 + ,) £ + l 



— y 



(1 



-^-0 



+ 2(1 + ^ log | 



(74) 



Having this expression for the moment Mi, we obtain the deflections of 
the plate by superposing expression (73) and expression (/), in which the 
constants of integration are given by expressions (i). 

By using the same method of superposition we can obtain also the 
solution for the case shown in Fig. 34, in which the plate is supported 
along the outer edge and carries a uniformly distributed load. In this 
case we use the solution obtained in the previous article for the plate 
without a hole at the center. Considering the section of this plate cut by 
the cylindrical surface of radius b and perpendicular to the plate, we find 
that along this section there act a shearing force Q = irqb 2 /2Trb = qb/2 
and a bending moment of the inten- 
sity [see Eq. (69)] 



-> 



Mr = il (3 + " )(a2 ~ 62) 



A 



<r a — ->■ 



22 



Fig. 34 



Hence to obtain the stresses and de- 
flections for the case shown in Fig. 34, 

we have to superpose on the stresses and deflections obtained for the plate 
without a hole the stresses and deflections produced by the bending 
moments and shearing forces shown in Fig. 35. These latter quantities 
are obtained from expressions (72), (73), (e), and (/), with due attention 
being given to the sign of the applied shears and moments. 

Several cases of practical importance 
are represented in Fig. 36. In all these 
cases the maximum stress is given by a 
^ formula of the type 



r 






^(3+v)(a z -b z ) 



Fig. 35 



= k 



qa l 



or 



"max 



kP 
h 2 



(75) 



depending on whether the applied load is uniformly distributed over the 
surface or concentrated along the edge. The numerical values of the 
factor k, calculated 1 for several values of the ratio a/b and for Poisson's 
ratio v = 0.3, are given in Table 3. 

1 The calculations for cases 1 to 8 inclusive were made by A. M. Wahl and G. Lobo, 
Trans. ASME, vol. 52, 1930. Further data concerning symmetrically loaded circular 
plates with and without a hole may be found in K. Beyer, "Die Statik im Stahlbe- 
tonbau," 2d ed., p. 652, Berlin, 1948. 



62 



THEORY OF PLATES AND SHELLS 



Cosel 



f^z^ 



w„ 



Cose 6 



ZZ 



r 




■^ 



=F^ 



W n 



Cose 2 



Cose 7 





1' V , 



Z2Z2 



W„ 



& 




Cose 3 



Cose 8 





§ 



VC 



§ 



3 




Cose 4 




Cose 9 



jft , »»max 




Cose 5 



Cose 10 




Fig. 36 

Table 3. Coefficients k and k x in Eqs. (75) and (76) for the Ten 

Cases Shown in Fig. 36 



a/b = 


1.25 


1.5 


2 


3 


4 


5 


Case 


A- 


*i 


k 


*i 


k 


*i 


k 


*i 


k 


ti 


k 


fci 


1 
2 
3 
4 
5 

6 

7 

8 

9 

10 


1.10 

0.66 

0.135 

0.122 

0.090 

0.115 
0.592 
0.227 
0.194 
0.105 


0.341 

0.202 

0.00231 

0.00343 

0.00077 

0.00129 

0.184 

0.00510 

0.00504 

0.00199 


1.26 

1.19 

0.410 

0.336 

0.273 

0.220 
0.976 
0.428 
0.320 
0.259 


0.519 

0.491 

0.0183 

0.0313 

0.0062 

0.0064 

0.414 

0.0249 

0.0242 

0.0139 


1.48 
2.04 
1.04 
0.74 
0.71 

0.405 
1.440 
0.753 
0.454 
0.480 


0.672 

0.902 

0.0938 

0.1250 

0.0329 

0.0237 

0.664 

0.0877 

0.0810 

0.0575 


1.88 
3.34 
2.15 
1.21 
1.54 

0.703 
1.880 
1.205 
0.673 
0.657 


0.734 
1.220 
0.293 
0.291 
0.110 

0.062 
0.824 
0.209 
0.172 
0.130 


2.17 
4.30 
2.99 
1.45 
2.23 

0.933 

2.08 

1.514 

1.021 

0.710 


0.724 
1.300 
0.448 
0.417 
0.179 

0.092 
0.830 
0.293 
0.217 
0.162 


2.34 
5.10 
3.69 
1.59 
2.80 

1.13 

2.19 

1.745 

1.305 

0.730 


0.704 
1.310 
0.564 
0.492 
0.234 

0.114 
0.813 
0.350 
0.238 
0.175 



SYMMETRICAL BENDING OF CIRCULAR PLATES 



63 



The maximum deflections in the same cases are given by formulas of 
the type 

Pa 2 

or w mRX = ki -t^ (76) 



w n 






Eh' 



Eh' 



The coefficients k\ are also given in Table 3. 

When the ratio a/b approaches unity, the values of the coefficients k and 
ki in Eqs. (75) and (76) can be obtained with sufficient accuracy by con- 
sidering a radial strip as a beam with end 
conditions and loading as in the actual plate. 
The effect of the moments M t on bending is 
then entirely neglected. 

18, Circular Plate Concentrically Loaded. 
We begin with the case of a simply supported f 
plate in which the load is uniformly distrib- 
uted along a circle of radius b (Fig. 37a). 
Dividing the plate into two parts as shown M| 

in Fig. 376 and c, it may be seen that the 
inner portion of the plate is in the condition 
of pure bending produced by the uniformly 

distributed moments M i and that the outer part is bent by the moments 
Mi and the shearing forces Qi. Denoting by P the total load applied, 
we find that 




*-» 



(a) 



The magnitude of the moment Mi is found from the condition of con- 
tinuity along the circle r = b, from which it follows that both portions 
of the plate have, at that circle, the same slope. Using Eqs. (72) and 
(j) of the preceding article, we find the slope for the inner boundary of 
the outer portion of the plate equal to 

(dw\ a 2 b 2 Mi (I 1 - v b_\ 

\dr) r = b D(l - p)(a 2 - b 2 ) \b + 1 + v a 2 ) 



+ 



Pb 
8ttD 



[ak.i-i- 1 -' 



l + r 



+ 



A*K I + $r^)] « 



The inner portion of the plate is bent to a spherical surface, the curvature 
of which is given by expression (46). Therefore the corresponding slope 
at the boundary is 

( dw \ _ _ Mib 
W) r = b ~ D(l + v) 



(c) 



64 THEORY OF PLATES AND SHELLS 

Equating expressions (b) and (c), we obtain 



Mi 



(1 _ v)P{a > - V) < 1 + y)Pl0g a 



8ira 5 



4ir 



(d) 



Substituting this expression for Mi in Eq. (73), we obtain deflections of 
the outer part of the plate due to the moments Mi. The deflections due 
to the forces Qi are obtained from Eq. '(/) of the preceding article. Add- 
ing together both these deflections, we obtain for the outer part of the 
plate 

--ra^- r!) ( ] + "2 tt7 °-^) + (i " + r °> log s] (77) 

Substituting r = 6 in this expression, we obtain the following deflection 
under the load : 



(w) 



r«& 



-£[«*-^+iff-:^+»*g 



(«) 



To find the deflections of the inner portion of the plate, we add to the 
deflection (e) the deflections due to pure bending of that portion of the 
plate. In this manner we obtain 



w 



(1 - v)P(a 2 - 6 2 ) 



+ 



p 

StD 

P 
SirD 



b 2 - r 2 
2D(1 + v) I 8xa 2 

(b 2 + r 2 ) log - + r 2 - 6 2 + (a 2 - r 2 ) 



(1 + v)P log - 

(X 

(3 + v)a 2 - (1 



v)¥ 



(6 2 + r 2 ) log - + (a 2 - 6 2 ) 



2(1 + v)a? 
(3 + v)a 2 - (1 - 



1 ~ v)r 2 l 
)a 2 J 



2(1 + „)<" ' (78) 

If the outer edge of the plate is built 
in, the deflections of the plate are ob- 
tained by superposing on the deflec- 
tions (77) and (78) the deflections 
M 2 produced by the bending moments 
Jlf 2 uniformly distributed along the 
outer edge of the plate (Fig. 38) and 
of such a magnitude that the slope of the deflection surface at that edge 
is equal to zero. From expression (77) the slope at the edge of a simply 
supported plate is 




/dw\ 

\dr/r=a 



a 2 



6 2 



4ttD 1 + v 



a 



(/) 



SYMMETRICAL BENDING OP CIRCULAR PLATES 

The slope produced by the moments M 2 is 

M 2 a 



/dw\ _ 
\dr/ r=sa 



D(l + v) 
Equating the sum of expressions (/) and (g) to zero, we obtain 

p a 2 _ hi 

4ir a 2 
Deflections produced by this moment are 

M 2 r 2 - a 2 P 



65 



(g) 



w = 



a 2 - 6 2 



£(1 + v) 2 



8ttZ)(1 + v) a 2 



(r 2 - a 2 ) 



W 



Adding these deflections to the deflections (77) and (78) we obtain for the 
outer portion of a plate with a built-in edge 



w = 



SwD I 

and for the inner portion, 
P 



P '" (a2 " r2) ^2^ + (52 + r2) l0g h] 



(79) 



10 



8tt£> 
P 

87tD 



(6 2 + r 2 ) log - + r 2 - fe 2 + 



(a 2 - r 2 ){a* + b 2 )1 
2a 2 I 



(6 2 -f r 2 ) log - + —255- 



b 2 ) 



(80) 



Having the deflections for the case of a load uniformly distributed 
along a concentric circle, any case of bending of a circular plate sym- 
metrically loaded with respect to the center can be solved by using the 
method of superposition. Let us consider, for example, the case in which 
the load is uniformly distributed over the inner portion of the plate 



A" 






; -3 l*db 

H- ^ 

Fig. 39 

bounded by a circle of radius c (Fig. 39). Expression (77) is used to 
obtain the deflection at any point of the unloaded portion of the plate 
(a > r >c). The deflection produced by an elementary loading dis- 
tributed over a ring surface of radius b and width db (see Fig. 39) is 
obtained by substituting P = 2irbq db in that expression, where q is the 
intensity of the uniform load. Integrating the expression thus obtained 
with respect to 6, we obtain the deflection 



66 THEORY OF PLATES AND SHELLS 



+ b ,„„. (1 -v)(a*-r>) 



2 [ log a 2(1 + *)a 2 



6d6 



87) 



<?c 4 |\ r _ 1 — y a 2 — r 2 1 
+ I6D L g a 2(1 + v) a 2 J 



or, denoting the total load irc 2 q by P, 



+'hi-5^ a -^]l (81) 



Expression (78) is used to obtain the deflection at the center. Substi- 
tuting r = and P = 2wbq db in this expression and integrating, we find 

where P = 7rc 2 g. 

The maximum bending moment is at the center and is found by using 
expression (d). Substituting 2irbqdb for P in this expression and inte- 
grating, we find 

_|[ (1 + F)log 2 + 1 _iL^] (83) 

where, as before, P denotes the total load irc 2 q* 

Expression (81) is used to obtain the bending moments M r and M t at 
any point of the unloaded outer portion of the plate. Substituting this 
expression in the general formulas (52) and (53), we find 

^ = ^■0^0^(1-1) 

-.-Sla + ^^H-*^ 5 ^) (85) 

* This expression applies only when c is at least several times the thickness h. The 
case of a very small c is discussed in Art. 19. 



SYMMETRICAL BENDING OF CIRCULAR PLATES 67 

The maximum values of these moments are obtained at the circle r = c, 
where 

M r = y ^ r V ±L loo; 1 + iL_2^__J-2 ( 86 ) 

47r r? l()7ra- 



71/. = -£- 

47T 



(1 + v) 1<>K*+ 1 - v 



(I - f)P(o» + c») . 

"" ' "lew" (8/) 



The same method of calculating deflections and moments can be used also 
for any kind of symmetrical loading of a circular plate. 

The deflection at the center of the plate can easily be calculated also for 
any kind of unsymmetrical loading by using the following consideration. 

Owing to the complete symmetry of the plate and of its boundary con- 
ditions, the deflection produced at its center by an isolated load P depends 
only on the magnitude of the load and on its radial distance from the 
center. This deflection remains unchanged if the load P is moved to 
another position provided the radial distance of the load from the center 
remains the same. The deflection remains unchanged also if the load P 
is replaced by several loads the sum of which is equal to P and the radial 
distances of which are the same as that of the load P. From this it 
follows that in calculating the deflection of the plate at the center we can 
replace an isolated load P by a load P uniformly distributed along a circle 
the radius of which is equal to the radial distance of the isolated load. 
For the load uniformly distributed along a circle of radius b the deflection 
at the center of a plate supported at the edges is given by Eq. (78) and is 



(W)r«0 



8tt/> 



^±JL. („• - V) - 6« log |] (0 



This formula gives the deflection at the center of the plate produced by 
an isolated load P at a distance b from the center of the plate. Having 
this formula the deflection at the center for any other kind of loading 
can be obtained by using the method of superposition. 1 It should be 
noted that the deflections and stresses in a circular plate with or without 
a hole can be efficiently reduced by reinforcing the plate with either con- 
centric 2 or radial ribs. In the latter case, however, the stress distribution 
is no longer symmetrical with respect to the center of the plate. 

19. Circular Plate Loaded at the Center. The solution for a concen- 
trated load acting at the center of the plate can be obtained from the 

1 This method of calculating deflections at the center of the plate was indicated by 
Saint Venant in his translation of the "Theorie de l'elasticite des corps sondes," by 
Clebsch, p. 363, Paris, 1883. The result (i) can also be obtained by applying Max- 
well's reciprocal theorem to the circular plate. 

2 This case is discussed by W. A. Nash, J. Appl. Mechanics, vol. 15, p. 25, 1948. 
See also C. B. Biezeno and R. Gramme], "Teehnische Dynamik," 2d ed., vol. 1, p. 497, 
1953. 



68 



THEORY OF PLATES AND SHELLS 



discussion of the preceding article by assuming that the radius c of the 
circle within which the load is distributed becomes infinitely small, 
whereas the total load P remains finite. Using this assumption, we find 
that the maximum deflection at the center of a simply supported plate, 
by Eq. (82), is 

(3 + v)Pa 2 



Wm " 16V(1 + v)D 



(88) 



The deflection at any point of the plate at a distance r from the center, 
by Eq. (81), is 

P 



w = 



IQwD 



[f*->- 



r 2 ) + 2r 2 log - 



a 



(89) 



The bending moment for points with r > c may be found by omitting 
the terms in Eqs. (84) and (85) which contain c 2 . This gives 



Mr 

M t 



£<! + ') ^g 2 



4tt[ 



(1 + v) log - r + 1 

T 



-] 



(90) 
(91) 



To obtain formulas for a circular plate with clamped edges we differ- 
entiate Eq. (89) and find for the slope at the boundary of a simply sup- 
ported plate 




\dr) r = a 



Pa 



Fig. 40 



(a) 



4(1 + v)ttD 

M 2 The bending moments M 2 uniformly dis- 
tributed along the clamped edge (Fig. 40) 
produce a bending of the plate to a spher- 
ical surface the radius of which is given by Eq. (46), and the correspond- 
ing slope at the boundary is 

Using (a) and (6), the condition that the built-in edge does not rotate 

P 



gives 



(M r )r=a = M 2 = ~ 



47T 



(c) 



Deflections produced by moments M 2 by Eq. (h) of the preceding article 

are 

P(r* - a 2 ) 

8irD(l + v) 
Superposing these deflections on the deflections of a simply supported 



SYMMETRICAL BENDING OF CIRCULAR PLATES 



69 



plate in Eq. (89), we obtain the following expression for the deflections 
of a clamped plate loaded at the center : 



W = S l0g ^ + I^D (a2 - r2) 



(92) 



Adding Eq. (c) to Eqs. (90) and (91) for a simply supported plate, we 
obtain the following equations for the bending moment at any point not 
very close to the load: 



M t 



47r[ 



(1 + V) log 



a 



a 



(1 + y) log - r 



(93) 



(94) 



When r approaches zero, expressions (90), (91), (93), and (94) approach 
infinity and hence are not suitable for calculating the bending moments. 
Moreover, the assumptions that serve as the basis for the elementary 
theory of bending of circular plates do not hold near the point of appli- 
cation of a concentrated load. As the radius c of the circle over which 
P is distributed decreases, the intensity P/ttc 2 of the pressure increases 
till it can no longer be neglected in comparison with the bending stresses 
as is done in the elementary theory. Shearing stresses which are also 
disregarded in the simple theory likewise increase without limit as c 
approaches zero, since the cylindrical surface 2wch over which the total 
shear force P is distributed approaches zero. 

Discarding the assumptions on which the elementary theory is based, we may obtain 
the stress distribution near the point of application of the load by considering that 
portion of the plate as a body all three dimensions of which are of the same order of 
magnitude. To do this imagine the central 
loaded portion separated from the rest of the 
plate by a cylindrical surface whose radius b is 
several times as large as the thickness h of the 
plate, as shown in Fig. 41. It may be assumed 
that the elementary theory of bending is accur- 
ate enough at a distance b from the point of 
application of the load P and that the corre- 
sponding stresses may be calculated by means 
of Eq. (90). The problem of stress distribu- 
tion near the center of the plate is thus reduced to the problem of a symmetrical 
stress distribution in a circular cylinder of height h and radius b acted upon by a 
load P distributed over a small circle of radius c and by reactions along the lateral 
boundary. 1 The solution of this problem shows that the maximum compressive 

1 Several examples of symmetrical stress distribution are discussed in S. Timo- 
shenko and J. N. Goodier, " Theory of Elasticity," 2d ed., p. 384, 1951. The case 
shown in Fig. 41 was studied by A. Nadai (see his book "Elastische Platten," p. 308) 
and also by S. Woinowsky-Krieger (see his paper in Ingr.-Arch., vol. 4, p. 305, 1933). 
The results given here are from the latter paper. 




[B 

H b -4*- 

Fig. 41 



-j 



70 



THEORY OF PLATES AND SHELLS 



stress at the center A of the upper face of the plate can be expressed by the follow- 
ing approximate formula: 1 



o> 



P fl +2v , 1 



(95) 



in which <n is the value of the compressive bending stress 2 obtained from the approxi- 
mate theory, say, by using Eq. (83) for the case of a simply supported plate, and a is a 
numerical factor depending on 2c/h, the ratio of the diameter of the loaded area to the 

















0.4 
km 


























a 


0.2 




























U.I 

















0.5 1.0 



1.5 2.0 
— *"2c 
h 



2.5 3.0 



Fig. 42 

thickness of the plate. Several values of this factor are given in Table 4. Its varia- 
tion with the ratio 2c/h is shown also in Fig. 42. When c approaches zero, the stress 
calculated by Eq. (95) approaches infinity. 

Table 4. Values of Factor a in Eq. (95) 



2c/h = 


0.10 


0.25 


0.50 


0.75 


1.00 


1.50 


2.00 


2.50 


a = 


0.0106 


0.0466 


0.1234 


0.200 


0.263 


0.348 


0.386 


0.398 



The maximum tensile stress occurs at B, the center of the lower surface of the plate 
(Fig. 41). When c is very small, i.e., for a strong load concentration, this tensile 
stress is practically independent of the ratio 2c/h and for a simply supported plate is 
given by the following approximate formula: 3 



tfmax 



= j£ (1 + p) ( 0.485 log £ + 0.52 j + 0.48 



(96) 



in which a is the outer radius. 

To obtain the compressive stresses <r r and at at the center of the upper surface of a 
clamped plate, we must decrease the value of the compressive stress <n in Eq. (95) 
by an amount equal to 

Air h* 2wh* } 

1 When c is very small, the compressive stress P/wc 2 becomes larger than the value 
of or m ax given by Eq. (95) (see Fig. 43). 

2 This quantity should be taken with negative sign in Eq. (95). 

3 See Woinowsky-Krieger, op. cil. 



SYMMETRICAL BENDING OF CIRCULAR PLATES 



71 



on account of the action of the moments M 2 = -P/4*-. The maximum tensile 
stress at the center of the lower surface of a clamped plate for a strong concentration 
of the load (c = 0) is found by subtracting Eq. (d) from Eq. (96). This stress is 



Cmax 



-£<• 



+ y) 



( 0.485 log ^ + 0.52 j 



(97) 



The stress distribution across a thick circular plate (h/a = 0.4) with built-in edges 
is shown in Fig. 43. These stresses are calculated for c = 0.1a and v = 0.3. For this 
case the maximum compressive stress <r t normal to the surface of the plate is larger than 
the maximum compressive stress in bending given by Eq. (95). The maximum 



r»0 



-100 "92.3 




Fig. 43 

tensile stress is smaller than the tensile stress given by the elementary theory of bend- 
ing. The value of the latter across the thickness of the plate is shown in the figure 
by the dashed line. It was calculated from the equation for bending moment 



M„ 



4*r l 



, a (1 - v)c 2 
d + log- ^ 



] 



(98) 



obtained by adding the moment M 2 = — P/4ir to Eq. (83). 

In determining the safe dimensions of a circular plate loaded at the center, we can 
usually limit our investigations to the calculation of the maximum tensile bending" 
stresses at the bottom of the plate by means of Eqs. (96) and (97). Although the 
compressive stresses at the top of the plate may be many times as large as the tensile 
stresses at the bottom in the case of a strong concentration of the load, they do not 
represent a direct danger because of their highly localized character. The local 
yielding in the case of a ductile material will not affect the deformation of the plate in 
general if the tensile stresses at the bottom of the plate remain within safe limits. 
The compressive strength of a brittle material is usually many times greater than its 
tensile strength, so that a plate of such a material will also be safe if the tensile stress 
at the bottom is within the limit of safety. 

The local disturbance produced by a concentrated load in the vicinity of its point of 
application must also be considered if we want an exact description of the deflection 
of the plate. This disturbance is mainly confined to a cylindrical region of radius 
several times h, and thus its effect on the total deflection becomes of practical impor- 
tance when the thickness of the plate is not very small compared with its radius. As 
an illustration there are shown in Fig. 44 the deflections of circular plates with built-in 
edges and a central concentrated load for which the ratio of thickness to radius h/a 



72 



THEORY OF PLATES AND SHELLS 



is 0.2, 0.4, and O.6. 1 The deflection given by the elementary theory [Eq. (92)] is 
shown by the dashed line. It may be seen that the discrepancy between the elementary 
theory and the exact solution diminishes rapidly as the ratio h/a diminishes. In the 
next article we shall show that this discrepancy is due principally to the effect of shear- 
ing forces which are entirely neglected in the elementary theory. 

20. Corrections to the Elementary Theory of Symmetrical Bending of 
Circular Plates- The relations (37) and (38) between bending moments 
and curvatures, which were derived for the case of pure bending, have 
been used as the basis for the solution of the various problems of sym- 
metrical bending of circular plates which have been discussed. The effect 
that shearing stresses and normal pressures on planes parallel to the sur- 
face of the plate have on bending has not been taken into account. Hence 




Fig. 44 

only the solution for a plate bent to a spherical surface and the solution 
for the annular plate loaded with moments uniformly distributed along 
the inner and outer boundaries (Fig. 31) are rigorous. In all other cases 
discussed, the formulas obtained are approximate, and their accuracy 
depends on the ratio of the thickness of the plate to its outer radius. 
More accurate formulas may be obtained by considering in an approxi- 
mate manner 2 the effect of shearing stresses and lateral pressures on 
deflections. 

Let us consider first a circular plate without a hole supported along its 
edge and uniformly loaded. The shearing force Q per unit length of arc 

1 The curves in Fig. 44 are the results of the exact solution of Woinowsky-Krieger, 
loc. cit. 

2 A rigorous theory of plates was originated by Saint Venant in his translation of 
Clebsch's "Th£orie de P&asticite des corps solides, ,, p. 337. A valuable criticism of 
this work is given in " History of the Theory of Elasticity/' by I. Todhunter and 
K. Pearson, vol. 2, part 1, p. 217. Further development of the theory is due to J. H. 
Michell, Proc. London Math. Soc, vol. 31, p. 100, 1900, and to A. E. H. Love, "Mathe- 
matical Theory of Elasticity," 4th ed., p. 465. A list of references on this subject is 
given by Woinowsky-Krieger, op. cit., p. 203. Some examples of rigorous theory are 
given in Art. 26 (see p. 98). 



SYMMETRICAL BENDING OF CIRCULAR PLATES 73 

along a circle of radius r is 

Q = W 

From the exact solution for plates whose thickness is not assumed to be 
small, 1 it is known that the shearing stresses r rz vary across the thickness 
of the plate according to the parabolic law in the same way as in beams of 
narrow rectangular cross section. Hence the maximum shearing stress is 
at the middle surface of the plate, and its magnitude is 

( __ 3 qr , ^ 

The corresponding shearing strain is 

dm __ __ 3 qr „ x 

IF ~ 2 2Gh K } 

where Wi is the additional deflection of the middle surface of the plate 
due to the shearing stress. By integration the deflections produced by 
the shearing stresses are found to be 



3 q 

Wi = 



2 4Gh 
and at the center of the plate, 



(a 2 - r 2 ) (c) 



The lateral pressure acting on the plate produces a positive curvature; 
convex downward, similar to that which occurs in a uniformly loaded 
beam. 2 The pressure q per unit area produces a radial elongation of 
vq/E at the upper surface of the plate. At the middle surface of the 
plate this elongation is vq/2E, and at the bottom of the plate it is zero. 
Assuming a straight-line relation to hold, an approximate value of the 
radius of curvature R can be found from the equation 



from which 



n _ 

2E 


h 
2R 


l 


vq 



2R 2hE 



and the negative deflection is 



w > - - 35 ( ° 2 - r!) " - 3& (a ' - r!) w 



1 Timoshenko and Goodier, op. cit., p. 351. 

2 See ibid., p. 43. 



74 THEORY OF PLATES AND SHELLS 

Adding Eqs. (c) and (e) to Eq. (67), a more exact expression for deflec- 
tion is found to be 



" = 64D (C 



At the center of the plate this becomes 

_ga^ / 5+ y 4 3 + » /t 2 \ m 

Wn " 64D \l + v "^ 3 1 - v 2 a 2 / U; 

The second term in Eq. (/) represents the correction for shearing stresses 
and lateral pressure. This correction is seen to be small when the ratio 
of the thickness of the plate to its radius is small. The value of this 
correction given by the exact solution is 1 

qa 4 2 8 + v + v 2 h 2 



64Z) 5 1 - v 2 a 2 



(g) 



For v = 0.3 the exact value is about 20 per cent less than that given by 
Eq. (/). 

In a uniformly loaded circular plate with clamped edges the negative 
deflection w 2 due to pressure cannot occur, and hence only the deflection 
Wi due to shear need be considered. Adding this deflection to Eq. (62), 
we obtain as a more accurate value of the deflection 



w = 6& [ (a2 ~ f2)2 + r=~v ( ° 2 ~ r2) ] w 



It is interesting to note that this coincides with the exact solution. 2 

Consider next the deflections produced by shearing stresses in the 
annular plate loaded with shearing forces uniformly distributed along 
the inner edge of the plate as shown in Fig. 32. The maximum shearing 
stress at a distance r from the center is 

3 P 

V rz/max 



2 2irrh 

where P denotes the total shear load. The corresponding shear strain is 8 

dwi _ _ 3 P , .v 

~W ~ 2 2~rM W 

Integrating, Ave obtain for the deflection produced by shear 

3 P . a Ph 2 . a ,~ 

Wl = I7W l0g r = Mr^ l0g r U) 

1 See Love, op. cit., p. 481. 

2 See ibid., p. 485. 

3 If the plate has no hole, the right-hand side of Eq. (i) should be multiplied by a 
factor (1 —>)/(! + v), in accordance with the result (0 given below. 



SYMMETRICAL BENDING OF CIRCULAR PLATES 



75 



This deflection must be added to Eq. (k) on page 60 to get a more 
accurate value of the deflection of the plate shown in Fig. 32. When 
the radius b of the hole is very small, the expression for the total deflec- 
tion becomes 



w = 



3 + v 



(a 2 - r 2 ) + r 2 log - 



8icD [2(1 + v) 
The deflection at the edge of the hole is 



+ 



PW 



a 



8tt(1 



*^max 



Pa 2 
StD 



3 + 



2(1 + v) 



+ 



1 h 



1 — v a 



T2 ] °g 



a 



IfB^r <*> 



(I) 



The second term in this expression represents the correction due to shear. 
It increases indefinitely as b approaches zero, as a consequence of our 
assumption that the load P is always finite. Thus when b approaches 
zero, the corresponding shearing stress and shearing strain become 
infinitely large. 

The term in Eq. (I) which represents the correction for shear cannot be applied to a 
plate without a hole. The correction for a plate without a hole may be expected to be 
somewhat smaller because of the wedging effect produced by the concentrated load P 
applied at the center of the upper surface of the plate. Imagine that the central 
portion of the plate is removed by means of a cylindrical'section of small radius b and 
that its action on the remainder of the plate is replaced by vertical shearing forces 
equivalent to P and by radial forces S representing the wedging effect of the load and 



S S 









fk 



i 



H---r --*+«— r -A 



TM- 



V-r- 



4T 



-r 



-j 



H2b 



Fig. 45 



(cO 



(b) 



Fkj. 4G 



distributed along the upper edge of the hole as shown in Fig. 45. It is evident that the 
latter forces produce stretching of the middle surface of the plate together with some 
deflection of the plate in the upward direction. This indicates that we must decrease 
the correction term in expression (k) to make it apply to a plate without a hole. To 
get an idea of the magnitude of the radial forces S, let us consider the plate under the 
two loading conditions shown in Fig. 46. In the first case the plate is compressed by 
two equal and opposite forces P acting along the axis of symmetry z. In the second 
case the plate is subjected to uniform compression in its plane by a pressure p uni- 
formly distributed over the cylindrical surface bounding the plate. As a result of 
lateral expansion these pressures produce an increase of the thickness of the plate by 
the amount 

E 



76 THEORY OF PLATES AND SHELLS 

We can now obtain from this expression the increase Ar in the radius r of the plate due 
to the action of the forces P (Fig. 46a) by applying the reciprocal theorem to the two 
conditions of loading shown in Fig. 46. This gives 

P Ah = 2wrhp Ar 

from which 

PAh 2v P 

Ar = = (rn) 

2irrhp E 2wr 

Let us compare this radial expansion with the radial expansion produced in a thick- 
walled cylinder by an internal pressure pi. If the inner radius b of the cylinder is very 
small compared with the outer radius r, the increase in the outer radius by Lamp's 
formula 1 is 

Ar = — (n) 

E r 

Comparing expressions (m) and (n), we conclude that the radial expansion which the 
forces P in Fig. 46a produce in the plate has the same magnitude as the radial expansion 
produced in a plate with a small cylindrical hole at the center (Fig. 45) by internal 
pressure pi whose magnitude is given by the equation 



2vP 

E2irr 


1 +v 
E 


Pib* 
r 




vP 




Pi — 


(1 +y) 


wb* 



From this we obtain 

(o) 

Returning to the case of one concentrated force at the center of the upper surface of 
the plate, the action of which is illustrated by Fig. 45, we conclude that the force S per 
unit length of the circumference of the hole must be equal to the pressure pih/2. 
Using the value of p t - from Eq. (o), we obtain 

*- > ph 



2(1 + p)irb* 



These forces applied in the upper plane of the plate produce upward deflections w\ 9 
the magnitude of which is found by substituting 

Sh vPh* 

Mi = 



2 4(1 + r)wb* 

in Eq. (73) and neglecting b 2 in comparison with a 2 . In this manner we obtain 

vPh 2 a 2 - r 2 vPh 2 , a 



Wi = — 



8tt(1 + v) 2 D a 2 4(1 - p 2 )itD ° r 



log - (p) 



Adding this to expression (k), we obtain the following more accurate formula for the 
deflection of a plate without a hole and carrying a load P concentrated at the center 
of the upper surface of the plate : 

1 See S. Timoshenko, " Strength of Materials," part II, 3d ed., p. 210, 1956. 



SYMMETRICAL BENDING OF CIRCULAR PLATES 77 



P T 3 + v r\ Ph 2 , a 

w = —^\ o/i i n (a2 "" r2) + r2 log "" + q n Z ^n log " 
ottD 2(1 + j>) a J 8tt(1 + y)Z> r 



vPh? a 2 -? 2 



8tt(1 + vYD 



This equation can be used to calculate the deflection of all points of the plate that 
are not very close to the point of application of the load. When r is of the same order 
of magnitude as the thickness of the plate, Eq. (q) is no longer applicable; and to 
obtain a satisfactory solution the central portion of the plate must be considered, as 
explained in the preceding article. We can get an approximate value of the deflection 
of this central portion considered as a plate of small radius b by adding the deflection 
due to local disturbance in stress distribution near the point of application of the load 
to the deflection given by the elementary theory. 1 The deflection due to local dis- 
turbance near the center is affected very little by the conditions at the edge of the 
plate and hence can be evaluated approximately by means of the curves in Fig. 44. 
The dashed-line curve in this figure is obtained by using Eq. (92). The additional 
deflections due to local stress disturbance are equal to the differences between the 
ordinates of the full lines and those of the dashed line. 

As an example, consider a plate the radius of the inner portion of which is b = 5/i. 
The deflection of the inner portion calculated from Eq. (92) and taken as unity in 
Fig. 44 is 

6i = = (5/i) 2 

16ttjD 16tt£> 

Using the curve h/a = 0.2 in Fig. 44, the additional deflection due to local stress 

disturbance is 

P 
5 2 = 0.215! = 0.21 — — (5A) 2 (r) 

loirD 

If we consider a plate for which b = 2.5h and use the curve for h/a = 0.4 in Fig. 44, 
we obtain 

52 = °- 81 vf^ < 2 ' 5/l ) 2 (5) 

IotD 

which differs only slightly from that given in expression (r) for b = 5h. It will be 
unsatisfactory to take b smaller than 2.5h, since for smaller radii the edge condition 
of the thick plate becomes of importance and the curves in Fig. 44, calculated for a 
built-in edge, may not be accurate enough for our case. 

Finally, to obtain the deflection of the plate under the load we calculate the deflec- 
tion by means of Eq. (q), putting r = in the first term and r = b = 2.5h in both 
other terms. To this deflection we add the deflection of the central portion of the 
plate due to the shear forces as given by expression (s). 

In the particular case of v = 0.3 the deflections of simply supported circular plates 
may also be obtained by a simple superposition of the curves plotted in Fig. 44,* with 
the deflection 

P(a* - r 2 ) 
8ttD(1 + v) 

1 In the case under consideration this deflection can be calculated by using the first 
term in expression (q) and substituting b for a. 
* Figure 44 was calculated for v = 0.3. 



78 THEORY OF PLATES AND SHELLS 

due to the pure bending by radial moments P/4x applied along the boundary of the 
plate. 

It should be noted also that, for small values of the ratio r/a, the effect of the 
shearing force P/2irr upon the deflection is represented mainly by the second term 
on the right-hand side of Eq. (q). To this term corresponds a slope 

dwx = _ 3 1 - v P (t) 

dr 2 ! 1 + v 2wrh.G 

Comparing this result with the expression (i), we conclude that the factor 

k = 00 

2 1 + *> 

if introduced into Eq. (i) instead of k = f , would give a more accurate value of the 
deformation due to shear in the case of a plate without a hole. 

All preceding considerations are applicable only to circular plates bent to a surface 
of revolution. A more general theory of bending taking into account the effect of the 
shear forces on the deformation of the plate will be given in Arts. 26 and 39. 



CHAPTER 4 

SMALL DEFLECTIONS OF LATERALLY LOADED PLAT'ES 



21. The Differential Equation of the Deflection Surface. We assume 
that the load acting on a plate is normal to its surface and that the 
deflections are small in comparison with the thickness of the plate (see 
Art. 13). At the boundary we assume that the edges of the plate are 
free to move in the plane of the plate; thus the reactive forces at the 
edges are normal to the plate. With these assumptions we can neglect 
any strain in the middle plane of the plate during bending. Taking., as 






k3I 



-dx 




*£& 



'Mx + -~ — dx 
9x 

M^ y+ ^dx 



~Q 



dx 

* + dx 



dx 



Fig. 47 



before (see Art. 10), the coordinate axes x and y in the middle plane of 
the plate and the z axis perpendicular to that plane, let us consider an 
clement cut out of the plate by two pairs of planes parallel to the xz and 
!/z planes, as shown in Fig. 47. In addition to the bending moments M x 
and My and the twisting moments M xu which were considered in the pure 
bending of a plate (see Art. 10), there are vertical shearing forces 1 acting 
on the sides of the element. The magnitudes of these shearing forces 
per unit length parallel to the y and x axes we denote by Q x and Q v , 
respectively, so that 



Q. = /. 



h/2 
-h/2 



x? dz 



Q» = J 



h/2 
-h/2 



dz 



(a) 



Since the moments and the shearing forces are functions of the coordi- 
nates x and y, we must, in discussing the conditions of equilibrium of the 
clement, take into consideration the small changes of these quantities 
when the coordinates x and y change by the small quantities dx and dy. 

1 There will be no horizontal shearing forces and no forces normal to the sides of the 
dement, since the strain of the middle plane of the plate is assumed negligible. 

70 



80 



THEORY OF PLATES AND SHELLS 



The middle plane of the element is represented in Fig. 48a and 6, and the 
directions in which the moments and forces are taken as positive are 
indicated. 

We must also consider the load distributed over the upper surface of 
the plate. The intensity of this load we denote by q } so that the load 
acting on the element 1 is q dx dy. 



'& 



<3M V s 



M y 



^M y> 



<2L £h /^Ta 



\M x + -r JL dx 



d Mux 
"M V x+"TT L dy 



M xy +-^dx 



"yx 1 



(a) 



<Jy 



,Q y 






Qy + -xf d v 



(b) 
Fig. 48 



'•.♦£*■ 



Projecting all the forces acting on the element onto the z axis we obtain 
the following equation of equilibrium: 



from which 



-~^ dx dy + -^ dy dx + qdxdy = 
eta dy 



dx dy 



(99) 



Taking moments of all the forces acting on the element with respect to 
the x axis, we obtain the equation of equilibrium 



xy dx dy -^ dy dx + Q y dxdy = 



dx 



dy 



(*) 



1 Since the stress component <r z is neglected, we actually are not able to apply the 
load on the upper or on the lower surface of the plate. Thus, every transverse single 
load considered in the thin-plate theory is merely a discontinuity in the magnitude of 
the shearing forces, which vary according to the parabolic law through the thickness 
of the plate. Likewise, the weight of the plate can be included in the load q without 
affecting the accuracy of the result. If the effect of the surface load becomes of 
special interest, thick-plate theory has to be used (see Art. 19). 



SMALL DEFLECTIONS OF LATERALLY LOADED PLATES 81 

The moment of the load q and the moment due to change in the force Q y 
are neglected in this equation, since they are small quantities of a higher 
order than those retained. After simplification, Eq. (b) becomes 

:»»_!«. + Q , = o (0 

dx ay 

In the same manner, by taking moments with respect to the y axis, we 
obtain 

*JjLz + ^ _ Q x = (d) 

dy dx 

Since there are no forces in the x and y directions and no moments 
with respect to the z axis, the three equations (99), (c), and (d) com- 
pletely define the equilibrium of the element. Let us eliminate the 
shearing forces Q x and Q y from these equations by determining them from 
Eqs. (c) and (d) and substituting into Eq. (99). In this manner we obtain 

d*M x d*M yx d*M y _ dhM X!l _ (e) 

dx n - "*■ dx dy "*" dy 2 dxdy q 

Observing that M yx = -M x „, by virtue of r xy = r yx , we finally represent 
the equation of equilibrium (e) in the following form: 

dm. aw 9 aw, = (100) 

dx 2 ^ dif- dxdy H 

To represent this equation in terms of the deflections w of the plate, 
we make the assumption here that expressions (41) and (43), developed 
for the case of pure bending, can be used also in the case of laterally 
loaded plates. This assumption is equivalent to neglecting the effect on 
bending of the shearing forces Q x and Q y and the compressive stress a z 
produced by the load q. We have already used such an assumption in 
the previous chapter and have seen that the errors in deflections obtained 
in this way are small provided the thickness of the plate is small in com- 
parison with the dimensions of the plate in its plane. An approximate 
theory of bending of thin elastic plates, taking into account the effect of 
shearing forces on the deformation, will be given in Art. 39, and several 
examples of exact solutions of bending problems of plates will be dis- 
cussed in Art. 26. 

Using x and y directions instead of n and t, which were used in Eqs. 
(41) and (43), we obtain 



D \w +v W) M " = - D W + v eW) (101) 



M xy = -M vx = D(l - v) ~ (102) 



dx dy 



82 THEORY OF PLATES AND SHELLS 

Substituting these expressions in Eq. (100), we obtain 1 

d*w dHv d 4 w _ q nor) 

This latter equation can also be written in the symbolic form 

AAw = ± (104) 

where Aw = -^j + j-^ (105) 

It is seen that the problem of bending of plates by a lateral load q 
reduces to the integration of Eq. (103). If, for a particular case, a solu- 
tion of this equation is found that satisfies the conditions at the bounda- 
ries of the plate, the bending and twisting moments can be calculated 
from Eqs. (101) and (102). The corresponding normal and shearing 
stresses are found from Eq. (44) and the expression 

_ 6M XII 

Equations (c) and (d) are used to determine the shearing forces Q x and 
Q y , from which 



n dM yx , dM x 
Hx ~~ dy ' ' dx " 


n d (d 2 w d 2 w\ 
dx\dx 2 + dy 2 ) 


(106) 


dM v dM xu 
V " ~ dy dx 


d /d 2 w d 2 w\ 
l) dy\dx* + dy 2 ) 


(107) 


or, using the symbolic form, 






<?* = -D-^(Aw) 


dy 


(108) 



The shearing stresses r xz and r uz can now be determined by assuming 
that they are distributed across the thickness of the plate according to 
the parabolic law. 2 Then 

\Txz)mn.x Q h V//-Vmax cy I. 

1 This equation was obtained by Lagrange in 1811, when he was examining the 
memoir presented to the French Academy of Science by Sophie Germain. The 
history of the development of this equation is given in I. Todhunter and K. Pearson, 
"History of the Theory of Elasticity/' vol. 1, pp. 147, 247, 348, and vol. 2, part 1, p. 
263. See also the note by Saint Venant to Art. 73 on page 689 of the French transla- 
tion of "Theorie de Pelasticite des corps solides," by Clebsch, Paris, 1883. 

2 It will be shown in Art. 26 that in certain cases this assumption is in agreement 
with the exact theory of bending of plates. 



SMALL DEFLECTIONS OF LATERALLY LOADED PLATES 83 

It is seen that the stresses in a plate can be calculated provided the 
deflection surface for a given load distribution and for given boundary 
conditions is determined by integration of Eq. (103). 

22. Boundary Conditions. We begin the discussion of boundary con- 
ditions with the case of a rectangular plate and assume that the x and 
y axes are taken parallel to the sides of the plate. 

Built-in Edge. If the edge of a plate is built in, the deflection along 
this edge is zero, and the tangent plane to the deflected middle surface 
along this edge coincides with the initial position of the middle plane of 
the plate. Assuming the built-in edge to be given by x = a, the bound- 
ary conditions are 

(w)*-« = (^) -= (109) 



Simply Supported Edge. If the edge x = a of the plate is simply sup- 
ported, the deflection w along this edge must be zero. At the same time 
this edge can rotate freely with respect to the edge line; i.e., there are no 
bending moments M x along this edge. This 
kind of support is represented in Fig. 49. The 
analytical expressions for the boundary condi- 
tions in this case are 



<">~-° (0+'3fX-- (u0) 




Fig. 49 



Observing that d 2 w/dy 2 must vanish together with w along the rectilinear 
edge x = a, we find that the second of the conditions (110) can be 
rewritten as d 2 w/dx 2 = or also Aw = 0. Equations (110) are there- 
fore equivalent to the equations 

(w) x=a = {Aw) x = a = (111) 

which do not involve Poisson's ratio v. 

Free Edge. If an edge of a plate, say the edge x = a (Fig. 50), is 
entirely free, it is natural to assume that along this edge there are no 
bending and twisting moments and also no vertical shearing forces, i.e., 
that 

(M x ) x ^ a = (M X!/ ) x = a = (<?,)**« = 

The boundary conditions for a free edge were expressed by Poisson 1 in 
this form. But later on, Kirchhoff 2 proved that three boundary con- 
ditions are too many and that two conditions are sufficient for the com- 
plete determination of the deflections w satisfying Eq. (103). He showed 

1 See the discussion of this subject in Todhunter and Pearson, op. cit., vol. 1, p. 250, 
.iikI in Saint Venant, loc. cit. 

-See J. Crellc, vol. 40, p. 51, 1850. 



84 



THEORY OF PLATES AND SHELLS 



also that the two requirements of Poisson dealing with the twisting 
moment M xy and with the shearing force Q x must be replaced by one 
boundary condition. The physical significance of this reduction in the 
number of boundary conditions has been explained by Kelvin and Tait. 1 
These authors point out that the bending of a plate will not be changed 
if the horizontal forces giving the twisting couple Mxy dy acting on an 
element of the length dy of the edge x = a are replaced by two vertical 
forces of magnitude M xy and dy apart, as shown in Fig. 50. Such a 
replacement does not change the magnitude of twisting moments and 
produces only local changes in the stress distribution at the edge of the 
plate, leaving the stress condition of the rest of the plate unchanged. 

We have already discussed a par- 

.Mxy 




ay 



X 



-Mxy 



ticular case of such a transforma- 
tion of the boundary force system 
in considering pure bending of a 
plate to an anticlastic surface (see 
Art. 11). Proceeding with the 
foregoinglreplacement of twisting 



Fig. 50 



-Mxy + 3M*y dy couples along the edge of the plate 
and considering two adjacent ele- 
ments of the edge (Fig. 50), we 
find that the distribution of twisting moments M xy is statically equiva- 
lent to a distribution of shearing forces of the intensity 



« -■ - (*£) 



x=*a 



Hence the joint requirement regarding twisting moment M xy and shear- 
ing force Q x along the free edge x = a becomes 



'-(*-*£)..- ° 



(a) 



Substituting for Q x and M xy their expressions (106) and (102), we finally 
obtain for a free edge x = a: 



[ 



dx 3 -*" K V) dx dy* 



1 =0 

Jx=a 



(112) 



The condition that bending moments along the free edge are zero requires 

(S + '3D_" (113> 



1 See "Treatise of Natural Philosophy," vol. 1, part 2, p. 188, 1883. Independ- 
ently the same question was explained by Boussinesq, /. Math., ser. 2, vol. 16, pp. 
125-274, 1871; ser. 3, vol. 5, pp. 329-344, Paris, 1879. 



SMALL DEFLECTIONS OF LATERALLY LOADED PLATES 



85 



Equations (112) and (113) represent the two necessary boundary con- 
ditions along the free edge x = a of the plate. 

Transforming the twisting couples as explained in the foregoing dis- 
cussion and as shown in Fig. 50, we obtain not only shearing forces Q' z dis- 
tributed along the edge x = a but 




(Mxy) x = a . 



y = o 



x = o»;y=b 



y G^-oarb CM ^-o.*«b 



Fig. 51 



also two concentrated forces at the 
ends of that edge, as indicated in 
Fig. 51. The magnitudes of these 
forces are equal to the magnitudes 
of the twisting couple 1 M xy at the 
corresponding corners of the plate. 
Making the analogous transforma- 
tion of twisting couples M yx along the edge y = b, we shall find that in 
this case again, in addition to the distributed shearing forces Q' y , there 
will be concentrated forces M yx at the corners. This indicates that a 
rectangular plate supported in some way along the edges and loaded 
laterally will usually produce not only reactions distributed along the 
boundary but also concentrated reactions at the corners. 

Regarding the directions of these concentrated reactions, a conclusion 
can be drawn if the general shape of the deflection surface is known. 
Take, for example, a uniformly loaded square plate simply supported 
along the edges. The general shape of the deflection surface is indicated 
in Fig. 52a by dashed lines representing the section of the middle surface 

of the plate by planes parallel to the xz 
and yz coordinate planes. Considering 
these lines, it may be seen that near the 
corner A the derivative dw/dx, repre- 
senting the slope of the deflection sur- 
face in the x direction, is negative and 
decreases numerically with increasing y. 
Hence d 2 w/dx dy is positive at the cor- 
ner A. From Eq. (102) we conclude 
that M zy is positive and M yx is negative 
at that corner. From this and from 
the directions of M xy and M yx in Fig. 
48a it follows that both concentrated forces, indicated at the point x = a, 
y = b in Fig. 51, have a downward direction. From symmetry we conclude 
also that the forces have the same magnitude and direction at all corners 
of the plate. Hence the conditions are as indicated in Fig. 526, in which 




B = 2{M xy ) x=a , y = a = 2D(1 



V) \dx dy) 



x=a,y=a 



1 The couple M xy is a moment per unit length and has the dimension of a force. 



86 



THEORY OF PLATES AND SHELLS 



It can be seen that, when a square plate is uniformly loaded, the 
corners in general have a tendency to rise, and this is prevented by the 
concentrated reactions at the corners, as indicated in the figure. 

Elastically Supported and Elasti- 
cally Built-in Edge. If the edges = a 
of a rectangular plate is rigidly joined 
to a supporting beam (Fig. 53), the 
deflection along this edge is not zero 
and is equal to the deflection of the 
beam. Also, rotation of the edge is 
equal to the twisting of the beam. 
Let B be the flexural and C the torsional rigidity of the beam. The pres- 
sure in the z direction transmitted from the plate to the supporting beam, 
from Eq. (a), is 




Fig. 53 



-'--(*-*£). 



D 



dx 



£ + e-'V 



dhvl 

dy* \x=a 



and the differential equation of the deflection curve of the beam is 



B 



ydytjx-a dx\_ 



d 2 W 
dx* 



+ (2-v) 



d 2 w 
dy* 



jx=a 



(114) 



This equation represents one of the two boundary conditions of the plate 
along the edge x = a. 

To obtain the second condition, the twisting of the beam should be 
considered. The angle of rotation 1 of any cross section of the beam is 
— (dw/dx) x = a , and the rate of change of this an- 
gle along the edge is 



_ / d 2 w \ 
\dx dy) x 



Hence the twisting moment in the beam is 
— C(d 2 w/dx dy)x-a. This moment varies along 
the edge, since the plate, rigidly connected with 
the beam, transmits continuously distributed 
twisting moments to the beam. The magni- 
tude of these applied moments per unit length 
is equal and opposite to the bending moments 

M x in the plate. Hence, from a consideration of the rotational equilib- 
rium of an element of the beam, we obtain 







) a 



1 The right-hand-screw rule is used for the sign of the angle. 



SMALL DEFLECTIONS OF LATERALLY LOADED PLATES 87 

or, substituting for M x its expression (101), 

_ d ( d*w \ n fd*w d*w\ ( . 

This is the second boundary condition at the edge x = a of the plate. 

In the case of a plate with a curvilinear boundary (Fig. 54), we take 
at a point A of the edge the coordinate axes in the direction of the 
tangent t and the normal n as shown in the figure. The bending and 
twisting moments at that point are 

za n dz M nt = - / ^ZT nt dz (b) 

— ft/2 J —ft/2 

Using for the stress components <s n and r„< the known expressions 1 

c„ = <x x cos 2 a -{- <r y sin 2 a -+- 2r xy sin a cos a 

r nt = t iv (cos 2 a — sin 2 a) + (<r„ — <x x ) sin a COS a 



(c) 



we can represent expressions (6) in the following form: 

M n = M x cos 2 a + M„ sin 2 a - 2M X1/ sin a cos a 
Mnt •■= M* y (cos 2 a - sin 2 a) + (Jlf, - itfj,) sin a cos a 

The shearing force Q n at point A of the boundary will be found from the 
equation of equilibrium of an element of the plate shown in Fig. 546, 
from which 

Q n ds = Qzdy - Q„ dx 
or Q n = Qx cos a + Q y sin a (d) 

Having expressions (c) and (d), the boundary condition in each particular 
case can be written without difficulty. 

If the curvilinear edge of the plate is built in, we have for such an edge 

w = — = (e) 

dn 

In the case of a simply supported edge we have 

w = ilf n = (/) 

Substituting for M n its expression from the first of equations (c) and 
using Eqs. (101) and (102), we can represent the boundary conditions (/) 
in terms of w and its derivatives. 

If the edge of a plate is free, the boundary conditions are 

M w = V n = Qn~^f==0 (g) 

1 The x and y directions are not the principal directions as in the case of pure bend- 
ing; hence the expressions for M n and M nt will be different from those given by Eqs. 
(39) and (40). 



88 THEORY OF PLATES AND SHELLS 

where the term —dM nt /ds is obtained in the manner shown in Fig. 50 
and represents the portion of the edge reaction which is due to the dis- 
tribution along the edge of the twisting moment M ni . Substituting 
expressions (c) and (d) for M n , M ni , and Q n and using Eqs. (101), (102), 
(106), and (107), we can represent boundary conditions (g) in the follow- 
ing form: 

vAw + (1 - v) ^cos 2 a w + Bm*a^s + sin 2a —J = 

cos a — Aw + sin a — Aw -\- (I — j>) — cos 2a ■ a (116) 

3a; 3?/ 35 L dx dy 

where, as before, 



, 1 . o /d 2 w d*w\\ 



d 2 w , d 2 w 

Another method of derivation of these conditions will be shown in the 
next article. 

23. Alternative Method of Derivation of the Boundary Conditions. The differential 
equation (104) of the deflection surface of a plate and the boundary conditions can be 
obtained by using the principle of virtual displacements together with the expression 
for the strain energy of a bent plate. 1 Since the effect of shearing stress on the deflec- 
tions was entirely neglected in the derivation of Eq. (104), the corresponding expres- 
sion for the strain energy will contain only terms depending on the action of bending 
and twisting moments as in the case of pure bending discussed in Art. 12. Using 
Eq. (48) we obtain for the strain energy in an infinitesimal element 

dV = - D {[ J - 2(1 - v) — - — - - I J \\ dxdy (a) 

2 \\dx* dy*J [ to d y \dxdyj J) 

The total strain energy of the plate is then obtained by integration as follows: 

^l»//|(S+5) , -*-"[S5-(=i)ll** tu " 

where the integration is extended over the entire surface of the plate. 

Applying the principle of virtual displacements, we assume that an infinitely small 
variation 8w of the deflections w of the plate is produced. Then the corresponding 
change in the strain energy of the plate must be equal to the work done by the external 
forces during the assumed virtual displacement. In calculating this work we must 
consider not only the lateral load q distributed over the surface of the plate but also 
the bending moments M n and transverse forces Q n — (dM n t/ds) distributed along the 
boundary of the plate. Hence the general equation, given by the principle of virtual 
displacements, is 

1 This is the method by which the boundary conditions were satisfactorily estab- 
lished for the first time; see G. Kirchhoff in J. Crelle, vol. 40, 1850, and also his 
Vorlesungen uber Mathematische Physik, Mechanik, p. 450, 1877. Lord Kelvin took 
an interest in KirchhofTs derivations and spoke with Helmholtz about them; see the 
biography of Kelvin by Sylvanus Thompson, vol. 1, p. 432. 



SMALL DEFLECTIONS OF LATERALLY LOADED PLATES 89 

6V = I q 8wdxdy - I M n — - ds + j IQu ) 8w ds (b) 

The first integral on the right-hand side of this equation represents the work of the 
lateral load during the displacement 8w. The second, extended along the boundary 
of the plate, represents the work of the bending moments due to the rotation d(8w)/dn 
of the edge of the plate. The minus sign follows from the directions chosen for M n and 
the normal n indicated in Fig. 54. The third integral represents the work of the 
transverse forces applied along the edge of the plate. 

In the calculation of the variation 8V of the strain energy of the plate we use certain 
transformations which will be shown in detail for the first term of expression (117). 
The small variation of this term is 

ff (d 2 w\ 7 J rt ff d*w d 2 8w J J 

S JJ \w) dxdy = 2 JJw-i^ dzdy 

J J \_dx\ax 2 dx ) dx z dx J 

ff f d I d 2 w d 8w\ d /d z w \ dho 1 , 
= 2 / / — ( J ( 8w J H -8w\dxdy (c) 

J J L^V^ 2 dx / dx\dX z ) dX A J 

In the first two terms after the last equality sign in expression (c) the double integra- 
tion can be replaced by simple integrals if we remember that for any function F of x 
and y the following formulas hold: 



ff dF f 

II — dx dy = / F cos a. ds 

ff dF f 

II — dx dy = / F sin a. ds 



id) 



In these expressions the simple integrals are extended along the boundary, and a. is the 
angle between the outer normal and the x axis, as shown in Fig. 54. Using the first 
of formulas (d), we can represent expression (c) as follows: 

ff (d*w\ 7 7 rt ff d 4 w J J , ft f (dhv d 8w d z w \ 

8 I I [ J dx dy = 2 / / 8w dx dy + 2 / I — 8w I cos a ds (e) 

J J \^V ^ J J dx' J \dx* dx dx* ) 

Advancing along the boundary in the direction shown in Fig. 54, we have 

d 8w d 8w dn d 8w ds d 8w d 8w . 
_- 1 ,.- cos a sin a 

dx dn dx ds dx dn ds 

With this transformation, expression (e) becomes 

5 //(S0 d * d y = 2 jj B Swdxdy 

f d 2 w (d 8w d 8w . \ J f d z w ^ J /jps 

+ 2 / I cos a sin ol I cos a ds — 2 / — - 8w cos a as (/) 

J dx 2 \ dn ds ) J dx z 

Integrating by parts, we have 

f dhv . d 8w 7 d 2 w . | f d (d 2 w . \ 

I sin a. cos ol ds = sin a cos a 8w — / —I — - sin a cos ot I 8w as 

J dx 2 ds dx 2 \ J Or \dx 2 J 



90 THEORY OF PLATES AND SHELLS 

The first term on the right-hand side of this expression is zero, since we are integrating 
along the closed boundary of the plate. Thus we obtain 

/dhv . d 8w J f d (dhv . \ e , 
sin a cos a as = — / — I sin a cos a I 8w as 
dx 2 ds J ds \dx 2 ) 

Substituting this result in Eq. (/), we finally obtain the variation of the first term in the 
expression for the strain energy in the following form : 

ff (dhv\ J , rt ff d A w , J , rt f d 2 w 9 d 8w , 
8 / / I J dx dy = 2 / / 5w dx dy + 2 / — - cos 2 a ds 

J J vw * 77 dx* * j dx 2 dn 

, n f I" a /d 2 ™ . \ a 3 w 1 . j , x 

+ 2 ; — I sin a cos a ] cos a \ dw as (g) 

J l ds \ d * 2 / dx * J 

Transforming in a similar manner the variations of the other terms of expression (117), 
we obtain 

ff fd 2 w\ 7 J „ ff d*w 7 7 , rt f a 2 ™ . a 5w , 

6 / / I 1 dx dy = 2 I -— 8w dx dy + 2 / — sin 2 a ds 

J J \dy 2 ) J J dy* J dy 2 dn 

« f T d f d * w . \ , ^ - 1 • j /rx 
— 2 J — I sin a cos a I H sin a I £10 as (fc) 

/Y a 2 w d 2 ^ ^ , „ ff d*w 
& I I ax ay = 2 / / 5w ax ay 

77 dx 2 ay 2 * 77 ax 2 ch/ 2 

Ka 2 ^ a 2 w . \ d 8w 7 f \ d z w . , a 3 w 
cos 2 a H sin 2 a I ds — / \ — - — sin a H cos a 
dy 2 dx 2 ) dn J \dx 2 dy dx dy 2 

d \(dhv d 2 w\ . 11 ■, ,.x 

H I 7 l sin a cos a i 5w? " s w 

^asL\5x 2 dy 2 ) J J 

ff ( dhv\ , , n ff d*w 

8 I I I J dx dy = 2 / / — - — - 5m; dx dy 

77 \dxdyj " ; 77 dx 2 dy 2 

/dhv . d 8w , /" I * I" 3*** / . . 
sin a cos a ds + / \ — (sin 2 a — cos 2 a) 
dxay dn 7 l ds Ldxdy 



a 3 w a 3 w . , , 

I cos a — sin a > 8w ds (j) 



dx dy 2 dx 2 dy 

"dy using these formulas the variation of the potential energy will be represented in the 
following form: 



8V >= D [ I AAw 8w dx dy 



<ss 

f r , x (d 2 w n rt dhv . , a 2 ™ . \ , i 

+ I (1 — ?) I cos 2 a + 2 — Sin a COS a H sin 2 a I + v Aw \ 

7 L \^ 2 dxd V d V / J 

M, , d \(dhv dhv\ . d 2 ™ ■ . N l 

+ / 1 (1 ' — v) — I I sin a cos a (cos 2 a — sin 2 a) 

7 I 5s [ \ 5x 2 dy 2 / dx ay "J 



a 5w 

an 



/a 3 ti; a 3 w; \ (d*w d*w \ . } \ 

— ( J cos a - ( ; H sina 6ti; ds 1 (118) 

\ ax 3 ax dy 2 ) \ dy* dx 2 dy) j / 

Substituting this expression in Eq. (6) and remembering that 8w and d(8w)/dn are 
arbitrary small quantities satisfying the boundary conditions, we conclude that Eq. (6) 



SMALL DEFLECTIONS OF LATERALLY LOADED PLATES 91 

will be satisfied only if the following three equations are satisfied: 

(DAAw - q) bw dxdy = (k) 

f f T fdho dhv . , d 2 w . 9 \ , a I 
/ \ D (1 — v) I cos 2 a + 2 sin a cos a H sin 2 a ] + v Aw \ 

J I L \^ 2 toa 2/ <^ 2 / J 

Id bw 
ds = (Z) 
dn 

[ ( \ d \ ( d 2 W d 2 w\ . d 2 W , „ -on! 

/ \ D \(1 — v) — \ [ 1 Sin a COS a (COS 2 a — Sin 2 a) 

J \ \ ds l\ dx2 w dxd y J 

(d*w d*w \ (d*w d*w \ . } ( dM lU \\ K , A , v 

- ( 1 ] cos a - ( J Sin a } - I Q n — J J bw ds = (m) 

\dx* dxdy 2 / \dy* dx 2 dy) j \ x ds // 

The first of these equations will be satisfied only if in every point of the middle surface 
of the plate we have 

DAAw — q = 

i.e., the differential equation (104) of the deflection surface of the plate. Equations 
(I) and (m) give the boundary conditions. 

If the plate is built in along the edge, bw and d(bw)/dn are zero along the edge; and 
Eqs. (I) and (m) are satisfied. In the case of a simply supported edge, bw = and 
M n = 0. Hence Eq. (m) is satisfied, and Eq. (I) will be satisfied if 

( d 2 w d 2 w . , d 2 w . o \ , _ , N 

(1 — r) I COS 2 a. + 2 sin « COS a H sin 2 a I + v Aw = (n) 

V ; \ dz 2 ax d?/ dy 2 ) 

In the particular case of a rectilinear edge parallel to the y axis, a = 0; and we obtain 
from Eq. (n) 

d 2 w d 2 w 

h „ = 

dx 2 dy 2 

as it should be for a simply supported edge. 

If the edge of a plate is entirely free, the quantity bw and d(bw)/dn in Eqs. (Z) and 
(m) are arbitrary; furthermore, M« = and Q» - (dM nt /ds) = 0. Hence, from 
Eqs. (Z) and (m), for a free edge we have 

( d 2 w d 2 w . , d 2 w . \ 

(1 — *0 I cos 2 a + 2 Sin a cos a + -— Sin 2 a J + v Aw = 

V J \ dx 2 dx dy dy 2 / 

x d \ (d 2 W d 2 w\ . d 2 W . 1 

(1 — v) — 1 I sin « cos a (COS 2 a — Sin 2 a) 

v ^sLyaa; 2 dy 2 ) dxdy J 

( d z w d*w \ (d z w , d*w \ . 
— [ 1 ) cos a — I 1 — J sin a = 

\dx z dxdy 2 / \dy z dx 2 dy/ 

These conditions are in agreement with Eqs. (116) which were obtained previously 

(see page 88). In the particular case of a free rectilinear edge parallel to the y axis, 

a = 0, and Ave obtain 

d 2 w d 2 w 

h v = 

dx 2 dy 2 

d z W d z W 

— + (2 - v) = 

dx z dx dy 2 

These equations coincide with Eqs. (112) and (113) obtained previously. 



92 THEORY OF PLATES AND SHELLS 

In the case when given moments M n and transverse forces Q n — (dM n t/ds) are dis- 
tributed along the edge of a plate, the corresponding boundary conditions again can 
be easily obtained by using Eqs. (I) and (ra). 

24. Reduction of the Problem of Bending of a Plate to That of Deflec- 
tion of a Membrane. There are cases in which it is advantageous to 
replace the differential equation (103) of the fourth order developed for 
a plate by two equations of the second order which represent the deflec- 
tions of a membrane. 1 For this purpose we use form (104) of this 
equation : 

\dx 2 + dy 2 ) \dx 2 + dy 2 ) " D W 

and observe that by adding together the two expressions (101) for bend- 
ing moments (see page 81) we have 

*. + *,- -zxi-Mp + g) (» 

Introducing a new notation 

M = M X + M V = _ tdhv dhA 

1 + v \dx* ^ dy 2 ) U1W 

the two Eqs. (a) and (b) can be represented in the following form: 

aw aw 

% t = M (120) 

d 2 W d 2 W _ _ M ' 

dx 2 + dy 2 " "" T> 

Both these equations are of the same kind as that obtained for a uni- 
formly stretched and laterally loaded membrane. 2 

The solution of these equations is very much simplified in the case of 
a simply supported plate of polygonal shape, in which case along each 
rectilinear portion of the boundary we have d 2 w/ds 2 = since w = at 
the boundary. Observing that M n = at a simply supported edge, we 
conclude also that d 2 w/dn 2 = at the boundary. Hence we have [see 
Eq. (34)] 

ds 2 + dn 2 dx 2 + dy 2 ~ D W 

at the boundary in accordance with the second of the equations (111). 
It is seen that the solution of the plate problem reduces in this case to 
the integration of the two equations (120) in succession. We begin with 

1 This method of investigating the bending of plates was introduced by H. Marcus 
in his book "Die Theorie elastischer Gewebe," 2d ed., p. 12, Berlin, 1932. 

2 See S. Timoshenko and J. N. Goodier, "Theory of Elasticity," 2d ed., p. 269, 1951. 



SMALL DEFLECTIONS OF LATERALLY LOADED PLATES 93 

the first of these equations and find a solution satisfying the condition 
M = at the boundary. 1 Substituting this solution in the second equa- 
tion and integrating it, we find the deflections w. Both problems are of 
the same kind as the problem of the deflection of a uniformly stretched 
and laterally loaded membrane having zero deflection at the boundary. 
This latter problem is much simpler than the plate problem, and it can 
always be solved with sufficient accuracy by using an approximate 
method of integration such as Ritz's or the method of finite differences. 
Some examples of the application of these latter methods will be dis- 
cussed later (see Arts. 80 and 83). Several applications of Ritz's method 
are given in discussing torsional problems. 2 

A simply supported plate of polygonal shape, bent by moments M n 
uniformly distributed along the boundary, is another simple case of the 
application of Eqs. (120). Equations (120) in such a case become 

d 2 M d 2 M = Q 

to* W (121) 

d 2 w d 2 w M 

dx 2 + dy 2 D 



Along a rectilinear edge we have again d 2 w/ds 2 = 0. Hence 

and we have at the boundary 

d*wd*w_dhv = _Mn == _ M 
dx 2 + ~dy 2 " dn 2 ' D D 

This boundary condition and the first of the equations (121) will be 
satisfied if we take for the quantity M the constant value M = M n 
at all points of the plate, which means that the sum of the bending 
moments M x and M y remains constant over the entire surface of the 
plate. The deflections of the plate will then be found from the second 
of the equations (121), 3 which becomes 

d 2 w d 2 w _ _Mn , d) 

dx 2 + dy 2 D 

It may be concluded from this that, in the case of bending of a simply 
supported polygonal plate by moments M n uniformly distributed along 
the boundary, the deflection surface of the plate is the same as that of 

i Note that if the plate is not of a polygonal shape, M generally does not vanish at 
the boundary when M n = 0. 

2 See Timoshenko and Goodier, op. cit., p. 280. 

3 This was shown first by S. Woinowsky-Krieger, Ingr.-Arch., vol. 4, p. 254, 1933. 



94 



THEORY OF PLATES AND SHELLS 



a uniformly stretched membrane with a uniformly distributed load. 
There are many cases for which the solutions of the membrane problem 
are known. These can be immediately applied in discussing the corre- 
sponding plate problems. 

Take, for example, a simply sup- 
ported equilateral triangular plate 
(Fig. 55) bent by moments M n 
uniformly distributed along the 
boundary. The deflection surface 
of the plate is the same as that of 
a uniformly stretched and uni- 
formly loaded membrane. The 
latter can be easily obtained ex- 
perimentally by stretching a soap 
film on the triangular boundary 
and loading it uniformly by air 
pressure. 1 
The analytical expression of the deflection surface is also comparatively 
simple in this case. We take the product of the left-hand sides of the 
equations of the three sides of the triangle: 




Fig. 55 



( 



, a\ / x 2a 



K x _ 2a \ 

V3 V 3 Vs) 

x 3 — Zy 2 x a(x 2 + y 2 ) 



+ 



4a 3 
3-27 



This expression evidently becomes zero at the boundary. Hence the 
boundary condition w = for the membrane is satisfied if we take for 
deflections the expression 



w 



-*P 



: 3 - Sy 2 x a(x 2 + y 2 ) 



3 



+ 



4a 8 1 
3 • 27j 



(«) 



where Af is a constant factor the magnitude of which we choose in such a 
manner as to satisfy Eq. (d) . In this way we obtain the required solution : 



w 



- M * r r i_ -. 

~ 4aD L ' 



Zy*x =- a(x* + y 2 ) + 



H 



(/) 



Substituting x — y — in this expression, we obtain the deflection at the 
centroid of the triangle 

M„a 2 



Wo 



27D 



(9) 



1 Such experiments are used in solving torsional problems; see Timoshenko and 
Goodier, op. cit., p. 289. 



SMALL DEFLECTIONS OF LATERALLY LOADED PLATES 95 

The expressions for the bending and twisting moments, from Eqs. (101) 
and (102), are 

M n 



M x = 
M = Mi 

1X1 y i) 

Li 



l + ,-(l-,)| 

1 + v + (1 - v) ^1 (h) 



m - - 3(1 - v)M n y 

Mxu ~ 2a 

Shearing forces, from Eqs. (106) and (107), are 

Along the boundary, from Eq. (d) of Art. 22, the shearing force Q n = 0, 
and the bending moment is equal to M„. The twisting moment along 
the side BC (Fig. 55) from Eqs. (c) of Art. 22 is 

3(1 - v)M n . nz , 

M nt = — — -^ — (y ~ V 3 X) 

The vertical reactions acting on the plate along the side BC (Fig. 55) are 
T/ n dM nt 3(1 - v) M ({) 

From symmetry we conclude that the same uniformly distributed reac- 
tions also act along the two other sides of the plate. These forces are 
balanced by the concentrated reactions at the corners of the triangular 
plate, the magnitude of which can be found as explained on page 85 and 
is equal to 

R = 2(M„,)x-ia >w -o = (1 - v) V3 M» 0') 

The distribution of the reactive forces along the boundary is shown in 
Fig. 556. The maximum bending stresses are at the corners and act on 
the planes bisecting the angles. The magnitude of the corresponding 
bending moment, from Eqs. (h), is 

(M v )_ = (ar„w = Mn(3 9 " v) ( fc ) 

This method of determining the bending of simply supported polygonal 
plates by moments uniformly distributed along the boundary can be 
applied to the calculation of the thermal stresses produced in such plates 
by nonuniform heating. In discussing thermal stresses in clamped plates, 
it was shown in Art. 14 [Eq. (&)] that nonuniform heating produces uni- 
formly distributed bending moments along the boundary of the plate 
which prevent any bending of the plate. The magnitude of these 



96 
moments is 1 



THEORY OF PLATES AND SHELLS 



M n = 



atD(l + v) 
h 



(I) 



To obtain thermal stresses in the case of a simply supported plate we 
need only to superpose on the stresses produced in pure bending by the 
moments (Z) the stresses that are produced in a plate with simply sup- 
ported edges by the bending moments — atD(l + v)/h uniformly dis- 
tributed along the boundary. The solution of the latter problem, as was 
already explained, can be obtained without much difficulty in the case of 
a plate of polygonal shape. 2 



*R=~atEh* *R 




Fig. 56 

Take again, as an example, the equilateral triangular plate. If the 
edges of the plate are clamped, the bending moments due to nonuniform 
heating are 

M' m = M' y = atD{ \+ v) (m) 

To find the bending moments M x and M y for a simply supported plate 
we must superpose on the moments (ra) the moments that will be obtained 
from Eqs. (h) by letting M n = — atD{\ + v)/h. In this way we finally 
obtain 



M x = 



M v = 



M xv = 77 



■xy 



atD(l + v) 
h 


atD(l + v) 
2h 


atD(l + v) 

h 


atD(l + v) 
2h 


1 atEh 2 y 
8 a 





1 + v - (1 



»t] 



a 



24 



+ v + (1 - v) ^ 



a 



tEh'' 



24 



0-1) 

( x -t) 



1 It is assumed that the upper surface of the plate is kept at a higher temperature 
than the lower one and that the plate thus has the tendency to bend convexly upward. 

2 See dissertation by J. L. Maulbetsch, J. Appl. Mechanics, vol. 2, p. 141, 1935. 



SMALL DEFLECTIONS OF LATERALLY LOADED PLATES 97 

The reactive forces can now be obtained from Eqs. (i) and (j) by substi- 
tution of M n = —atDil + v)/h. Hence we find 

v _ n dMnt - atEh2 7? - - ^ atEh2 
Vn - U» 35 - 8a ^ 12 

The results obtained for moments and reactive forces due to nonuniform 
heating are represented in Fig. 56a and 6, respectively. 

26. Effect of Elastic Constants on the Magnitude of Bending Moments. It is seen 
from Eqs. (101) and (102) that the magnitude of the bending and twisting moments 
in a plate is considerably affected by the numerical value of Poisson's ratio v. On the 
other hand, it can be easily shown that in the case of a transverse load the magnitude 
of the quantity Dw is independent of both constants E and v if the plate is either 
simply supported at rectilinear edges or clamped along some edges, whether rectilinear 
or not. 

Assuming such boundary conditions in any combination, let us consider the follow- 
ing problem. Some values of the bending moments M x and M v being given numeri- 
cally for an assumed numerical value of v, these moments must be computed for a new 
value, say *>', of the same elastic constant. Let M x and M y be the new values of the 
bending moments. Writing Eqs. (101) first for v ) then for *>', eliminating from them 
the curvatures d 2 w/dx 2 and d 2 w/dy 2 , and solving the resulting equations for M x and 
M y , we obtain 

1 [(1 ~ w')M x + („' - v)M y ) 

(122) 
[(1 - vv')M v + {v' - v)M x ] 

Thus M x and M y can be readily calculated if M x and M y are known. 

If the constant v is implied in some of the given boundary conditions, as in the case 
of a free edge [Eq. (112)], Eqs. (122) do not hold any more. 

If the plate is elastically supported or elastically clamped, the moments also depend 
on the flexural rigidity D of the plate with respect to the stiffness of its restraint. 

The thermal stresses, finally, are affected not only by all the above-mentioned 
factors, but also by the absolute value of the rigidity D of the plate. 

Average values of v for some materials are given in Table 5. The last value of the 
table varies widely, depending on the age of the concrete, on the type of aggregate, 
and on other factors. 1 

Table 5. Average Values of Poisson's Ratio v 

Material v 

Steel 0.30 

Aluminum . 30 

Glass 0.25 

Concrete 0.15-0.25 

1 The German Code (DIN 4227) gives values of v which approximately can be 
expressed by v = V/c/350, f c being the compressive strength of concrete at 28 days 
in pounds per square inch. See also J. C. Simmons, Mag. of Concrete Research, vol. 
8, p. 39, 1956. 



X 




1 


— 


v' 2 


M' y 


= 




1 




1 


— 


v' 2 



98 THEORY OF PLATES AND SHELLS 

26. Exact Theory of Plates. The differential equation (103), which, together with 
the boundary conditions, defines the deflections of plates, was derived (see Art. 21) by 
neglecting the effect on bending of normal stresses u z and shearing stresses r xz and r yz . 
This means that in the derivation each thin layer of the plate parallel to the middle 
plane was considered to be in a state of plane stress in which only the stress components 
<r x , a y , and r xy may be different from zero. One of the simplest cases of this kind is that 
of pure bending. The deflection surface in this case is a second-degree function in x 
and y [see Eq. (c), Art. 11] that satisfies Eq. (103). The stress components <r x , <r y , 
and r xy are proportional to z and independent of x and y. 

There are other cases of bending in which a plane stress distribution takes place and 
Eq. (103) holds rigorously. Take, for example, a circular plate with a central circular 
hole bent by moments M r uniformly distributed along the boundary of the hole (Fig. 
57). Each thin layer of the plate cut out by two adjacent planes parallel to the middle 
plane is in the same stress condition as a thick-walled cylinder subjected to a uniform 
internal pressure or tension (Fig. 576). The sum <r r + <n of the two principal stresses 
is constant in such a case, Y and it can be concluded that the deformation of the layer in 
the z direction is also constant and does not interfere with the deformation of adjacent 
layers. Hence we have again a planar stress distribution, and Eq. (103) holds. 

Let us discuss now the general question regarding the shape of the deflection surface 
of a plate when bending results in a planar stress distribution. To answer this ques- 
tion it is necessary to consider the three differential equations of equilibrium together 
with the six compatibility conditions. If body forces are neglected, these equations 

2 



are 



dcr x dr xy dr x 



Sx dy dz 

d<r y dr xy dr yz 

1 1 = (a) 

dy dx dz 

d^ dr xt dr yz _ 

dz dx dy 



<M 



Ai<r x = 


1 d 2 


1 + v dx 2 


Aio-j, = 


l d 2 e 

1 + v dy 2 




l d 2 e 


Ai<r z = 


1 + v dz 2 


AlT X y = 


l 5 2 e 


1 + v dx dy 




l d 2 e 


&1T XZ = 


1 + v dx dz 


AiT... = 


1 d 2 6 



1 + v dy dz 
in which 

= <r x + <Ty + <Tz 

d 2 d 2 d 2 

and Ai = — - H H 

dx 2 dy 2 dz 2 

1 See Timoshenko and Goodier, op. cit., p. 60. 

2 See ibid., pp. 229, 232. 



SMALL DEFLECTIONS OF LATERALLY LOADED PLATES 99 

Adding Eqs. (ft), we find that 

d 2 e o 2 e d 2 e 

dx 2 dy 2 dz 2 V ' 

i.e., the sum of the three normal stress components represents a harmonic function. 
In the case of a planar stress r xz — r yz = a z — 0, and it can be concluded from the last 
two of the equations (c) and the last of the equations (ft) that dd/dz must be a constant, 
say /3. Hence the general expression for 6 in the case of planar stress is 

6 = O + $z (e) 

where 0o is a plane harmonic function, i.e., 

2 6u cJ 2 0o 

h = A0 O = 

dx 2 dy 2 

We see that in the case of planar stress the function consists of two parts: O 
independent of z and fiz proportional to z. The first part does not vary through the 
thickness of the plate. It depends on deformation of the plate in its own plane and 
can be omitted if we are interested only in bending of plates. Thus we can take in our 
further discussion 

= pz (/) 

Equations of equilibrium (a) will be satisfied in the case of a planar stress distribution 
if we take 

d 2 <p d 2 (p d 2 (p 

o\r = <r y — r xy = (g) 

dy 2 dx 2 dx dy 

where <p is the stress function. Let us consider now the general form of this function. 
Substituting expressions (</) in Eq. (/), we obtain 

d 2 <p d V 

— ~ H -= fa (h) 

dx 2 dy 2 

Furthermore, from the first of the equations (ft) we conclude that 

d 2 <p d 2 

Ai =0 or — - A\<p = 

dy 2 0y 2 

which, by using Eq. (h), can be put in the following form: 






(»') 



In the same manner, from the second and the third of the equations (b), we find 

dx 2 \dz 2 J dx dy \dz 2 J 

From Eqs. (i) and (j) it follows that d 2 <p/dz 2 is a linear function of x and y. This func- 
tion may be taken to be zero without affecting the magnitudes of the stress components 
given by expressions (g). In such a case the general expression of the stress function is 



100 THEORY OF PLATES AND SHELLS 

where <po is a plane harmonic function and <pi satisfies the equation 

dx 2 dy 2 

Since we are not interested in the deformations of plates in their plane, we can omit 
*pq in our further discussion and take as a general expression for the stress function 

<p = ip\Z \l) 

Substituting this in Eqs. (g), the stress components can now be calculated., and tho 
displacements can be found from the equations 



(m) 



du 1 , x dv 1 dw v 

— = — (<r x - v<T y ) — = — {<r v — v<t x ) — = — - ^x -t" <>W 

ax J5- dy E dz h 

du dv _ J. ^.^ = o ^+£^ = 

U» + da; ~ (? Txv dz dx a* a» 

For the displacements w perpendicular to the plate we obtain in this way 1 

P * + " 

™ * "~ 2£ (X2 + ^ + "^ + _ E~~ * l 

and the deflection of the middle surface of the plate is 

u> = - ^ (x 2 + 2/ 2 ) + — g- <Pi (») 

The corresponding stress components, from Eqs. (#) and (Z), are 

dy 2 dx 2 dxdy 

and the bending and twisting moments are 



(o) 



r*/2 ^3 3Vl r*/2 fc i a i w 

I. - / *** dz = — — — My = I ^ dz = 7^ "^7 
J -A/2 12 aj/* J-h/2 12 ax z 

/"fc/2 ^3 aVl 

J-h/2 12 dxdy 

For the curvatures and the twist of a plate, we find, from Eq. (n) 

dhv p l + v aVi d*w £ , l + " £Vi d*w = l + > • dVi r 

"a^i = ~ E + E ~dx* ~dy 2 E E dy* dxdy E dx dy 

from which, by using Eqs. (k) and (o), we obtain 



dhv dhv 1 - v 2 dVi _ 
'dx 2 dy 2 " E dy 2 


D 


dhv dhv 1 - v 2 d 2 ipi _ 
Ity 2 dx 2 "~ £ dx 2 ~~ 


My 

D 


a 2 !!? 1 + v d 2 <pi M xy 





(p) 



dx dy E dx dy (1 - v)D 

1 Several examples of calculating u, v, and w from Eqs. (?rc) are (jiven in ibid. 



SMALL DEFLECTIONS OF LATERALLY LOADED PLATES 



101 



From this analysis it may be concluded that, in the case of bending of plates resulting 
in a planar stress distribution, the deflections w [see Eq. (n)] rigorously satisfy Eq. 
(103) and also Eqs. (101) and (102) representing bending and twisting moments. If a 
solution of Eq. (k) is taken in the form of a function of the second degree in x and y, the 
deflection surface (n) is also of the second degree which represents the deflection for 




pure bending. Generally we can conclude, from Eq. (k) } that the deflection of the 
plate in the case of a planar stress distribution is the same as that of a uniformly 
stretched and uniformly loaded membrane. The plate shown in Fig. 57 represents a 
particular case of such bending, viz., the case for which the solution of Eq. (k), given 
in polar coordinates, is 

pi = Ar 2 + B log r + C 

where A, B, and C are constants that must be chosen so as to satisfy the boundary 
conditions. 

Plates of a polygonal shape simply supported and bent by moments uniformly 
distributed along the boundary (see Art. 24) represent another example of bending in 




Fig. 58 

which the deflection surface has a form satisfying Eq. (n), and Eqs. (101), (102), and 
(103) hold rigorously. In all these cases, as we may see from Eqs. (k) and (o), we have 






i.e., the sum of the bending moments in two perpendicular directions remains constant 
over the entire plate. 



102 THEOKY OF PLATES AND SHELLS 

Let us consider now the case in which bending of a plate results in a generalized 
planar stress distribution, i.e., one in which the normal stress component <r z is zero at 
all points of the plate and the shearing stress components r xz and r yz are zero on the 
surfaces z = ±h/2 of the plate. The deflection of a rectangular plate clamped along 
one edge and uniformly loaded along the opposite edge (Fig. 58) represents an example 
of such bending. From the theory of bending of rectangular beams we know that 
in this case a z = at all points of the plate and t xz is zero on the surfaces of the plate 
and varies along the depth of the plate according to the parabolic law 

6Q (h 2 \ 

Using again the general equations (a), (6), and (c) and proceeding as in the preceding 
case of a planar stress distribution, we find 1 that the general expression for the deflec- 
tion surface in this case has the form 



w 



= - [- + (i + ,)«] 



(?) 



in which <p is a planar harmonic function of x and y, and <pi satisfies the equation 

dVi #Vi 1 — v 

dx 2 dy 2 "~ 1 + v * 

It can be concluded that in this case again the differential equation (103) holds with 
q = 0. 

The equations for the bending and twisting moments and for the shearing forces in 
this case are 

/ A*nn fl*t#i\ R -4- v d 2 

Aw 
dy 2 



(123) 



/dhv d*w\ 8 + v ^ o a 2 

\dz 2 dy 2 J 40 a?/ : 

/a 2 it; dhvX 8 + j' a 2 / 

M y = -i) ( — + y — - ) + —£- Dh 2 — Ati; 

v \at/ 2 ax 2 / 40 ax 2 

a 2 ™ 8 + v a 2 

ilf ,„ = Z>(1 - v) + — — Dh 2 Aw 

dxdy 40 ax dy 

Q x = -D—Aw Q y = -D— Aiy 

dx dy 

Hence the expressions for the shearing forces coincide with expressions (108) given by 
the approximate theory, but the expressions for moments are different, the second 
terms of those expressions representing the effect of the shearing forces. 

These correction terms can be obtained in an elementary way by using the same 
reasoning as in the case of bending of beams. Considering the curvature in the xz 
plane, we can state that the total curvature is produced by two factors, the bending 
moments M XJ M y and the shearing force Q x . The curvature produced by the bending 

1 The rigorous solution for this case was given by Saint Venant; see his translation 
of Clebsch's "Theorie de Telasticite des corps solides/' p. 337. A general discussion 
of the rigorous theory of bending of plates was given by J. H. Michell, Proc. London 
Math. Soc, vol. 31, p. 100, 1900. See also A. E. H. Love, "The Mathematical Theory 
of Elasticity/' p. 473.. 1927. The results given in our further discussion are taken 
from the latter book. 



SMALL DEFLECTIONS OF LATERALLY LOADED PLATES 103 

moments is obtained by subtracting from the total curvature — dhv/dz 2 the portion 
-d(kQ x /hG)/dx produced by the shearing force. 1 Substituting 



\hGj 



d 
d 2 w 

dx 2 dx 

and -(d 2 w/dy 2 ) + d(kQ y /hG)/dy for -d 2 w/dx 2 and -dhv/dy 2 in Eqs. (101) and 

using the last two equations of the system (123), we find for the bending moments the 

expressions 

( d 2 w d 2 w\ kDh 2 d 2 

M x = -D [ h v M Aw 

\dx 2 dy 2 / 6 dy 2 

( d 2 w d 2 w\ kDh 2 d 2 

My = -D ( h v — r J + — : Aw 

y \dy 2 dx 2 / 6 dx 2 

These equations coincide with the first two equations of the system (123) if we take 

k _ 8 + v 
6 " 40 

For v = 0.3 this gives k = 1.245. 

From the theory of bending of beams we know that the correction due to the action 
of the shearing force is small and can be neglected if the depth h is small in comparison 
with the span of the beam. The same conclusion also holds in the case of plates. 

The exact expressions for stress components are 

Ez /dhu dhu\ E fh 2 z 2 - v n \ d 2 

— - Aw 



Txy — 



Txz = 



/dhv dhu\ E /h^z _ 2 - v \ 

ydx 2 V ~dy 2 ) 1 - v 2 \ 4 6 Z ) 

Ez /d 2 w d 2 w \ E (hH __ 2 - v \ 

~~ 1 - v 2 ydy 2 V dx 2 ) 1 - v 2 \ 4 6 Z ) 

(T-'-i-'') 



dy 2 

d 2 
Aw 

dx 2 



Ez d 2 w E fh 2 z 2 - v \ d 2 



(r) 



1 + v dx dy 1 - v 2 \ 4 6 ) dxdy 

E(h 2 - 4z 2 ) d E(h 2 - 4s 2 ) d 

AW T yz = — ~ AW <Tz = 

8(1 - v 2 ) dx v 8(1 - v 2 ) dy 



The second terms on the right-hand sides of the equations for <j x , a Uy and r xy are the 
corrections due to the effect of shearing forces on bending. It is seen that the stresses 
a Xy <r y , and r X y are no longer proportional to the distance z from the middle plane but 
contain a term proportional to 2 3 . Shearing stresses r xz and r yz vary according to the 
same parabolic law as for rectangular beams. In the case of a plane stress distribu- 
tion, Am; is a constant, and formulas (r) coincide with those given by the approximate 
theory. 

The problem of a uniformly loaded plate can also be treated rigorously in the same 
way. Thus it can be shown that the general expression for deflections in this case is 
obtained by adding to expression (</) the term 



1 q ( 2h 2 \ 

til (*' + *•)(«• + *- — r ) 



- J?; (x 2 + J/ 2 ) ( x 2 + j/ 2 - ) («) 

1 A; is a numerical factor that, in the case of beams depends on the shape of the crosi 
section. 



104 THEORY OF PLATES AND SHELLS 

which again satisfies Eq. (103) of the approximate theory. The equations for bending 
moments do not coincide with Eqs. (101) of the approximate theory but contain some 
additional correction terms. If the thickness of the plate is small in comparison with 
the other dimensions, these terms are small and can be neglected. 

In all previous cases general solutions of plate bending problems were discussed 
without considering the boundary conditions. There are also rigorous solutions of 
several problems in which boundary conditions are considered. 1 These solutions 
indicate that, provided the plate can be considered "thin," the customary theory is 
accurate enough for practical purposes except (1) in the vicinity of a highly con- 
centrated transverse load and (2) in narrow edge zones, especially near the corners of 
plates and around holes with a diameter of the order of magnitude of the plate thick- 
ness itself. 

In the first of these two cases the stress components <r z and the transverse shearing 
stresses must be considered equally important in their effect on the deformation of the 
plate. In obtaining the necessary correction to the stresses given by the approximate 
theory (see page 70) the boundary conditions can be eliminated from consideration. 
In such circumstances the thick-plate theory proves most convenient for the solution 
of the problem. 

In the second case the effect of the stress components <r z on the deformation becomes 
secondary as compared with the effect of the transverse shearing stresses r xt and r yz . 
Primarily taking into account this latter effect, several modified thin-plate theories 
have been developed recently (see Art. 39). These theories are better suited for the 
analysis of the stress distribution in the edge zone of the plates than the more rigorous 
thick-plate theory. 

1 In recent times the rigorous theory of plates has attracted the interest of engineers, 
and several important papers in this field have been published. We shall mention 
here the following: S. Woinowsky-Krieger, Ingr.-Arch., vol. 4, pp. 203 and 305, 1933. 
B. Galerkin, Compt. rend., vol. 190, p. 1047.; vol. 193, p. 568; vol. 194, p. 1440. G. D. 
Birkhoff, Phil. Mag., vol. 43, p. 953, 1922. C. A. Garabedian, Trans. Am. Math. Soc., 
vol. 25, p. 343, 1923; Compt. rend., vols. 178 (1924), 180 (1925), 186 (1928), 195 (1932). 
R. Archie Higdon and D. L. Holl, Duke Math. J., vol. 3, p. 18, 1937. A. C. Stevenson, 
Phil. Mag., ser. 7, vol. 33, p. 639, 1942; R. Ohlig, Ingr.-Arch., vol. 13, p. 155, 1942; 
I. N. Sneddon, Proc. Cambridge Phil Soc, vol. 42, p. 260, 1946;L.Leibenson, " Works," 
vol. 1, p. Ill, Moscow, 1951; H. Jung, Z. angew. Math. Mech., vol. 32, p. 57, 1952; 
E. Koppe, Z. angew. Math. Mech., vol. 37, p. 38, 1957. For thermal stresses see K. 
Marguerre, Z. angew. Math. Mech., vol. 15, p. 369, 1935; and I. S. Sokolnikoff and 
E. S. Sokolnikoff, Trans. Am. Math. Soc, vol. 45, p. 235, 1939. 



CHAPTER 5 



SIMPLY SUPPORTED RECTANGULAR PLATES 



27. Simply Supported Rectangular Plates under Sinusoidal Load. 

Taking the coordinate axes as shown in Fig. 59, we assume that the load 
distributed over the surface of the plate is given by the expression 



. WX . TTU 

q = O sin — sm -r- 
a b 



(a) 



in which q Q represents the intensity of the load at the center of the plate. 
The differential equation (103) for the deflection 
surface in this case becomes 



d 4 w 
dx* 



+ 2 



d A w 



+ 



d 4 w Go . ttx . ttv ,,. 
— - = -|r sin — sm -~ (6) 
D a b 




Fig. 59 



dx 2 dy 2 ' dy A 

The boundary conditions for simply supported 
edges are 

w = M x = for x = and a: = a 
w = My = for 2/ = and y = b 

Using expression (101) for bending moments and observing that, since 
w = at the edges, d 2 w/dx 2 = and d 2 w/dy 2 = for the edges parallel 
to the x and ?/ axes, respectively, we can represent the boundary condi- 
tions in the following form: 

d 2 w 



(1) u, 
(3) ti; 








(2) 



dx 2 







(4) ^ = ° 



for x = and a: = a 
for y = and y = b 



(c) 



It may be seen that all boundary conditions are satisfied if we take for 
deflections the expression 



„ . ttx . iry 
w = C sin — sin -— 
a b 



in which the constant C must be chosen so as to satisfy Eq. (6). 
tuting expression (d) into Eq. (6), we find 



Substi- 



-(h + k)' c -$ 



105 



106 THEORY OF PLATES AND SHELLS 

and we conclude that the deflection surface satisfying Eq. (6) and bound- 
ary conditions (c) is 

w = T" 2 ^2 sm — sm "IT v«) 

/l l\ a b 

Having this expression and using Eqs. (101) and (102), we find 

/ \1 ( -5 + k5 1 sin — sin IT 

,/i , iV\° 6 v a 6 (/) 

go(l - 

-*(p+p)«* 

It is seen that the maximum deflection and the maximum bending 
moments are at the center of the plate. Substituting x = a/2, y = 6/2 
in Eqs. (e) and (/), we obtain 

w— 7 9 ° v (124) 



M„ = 



,, go(l — *') tx 7r/y 

M™ = — j-^ r^ — cos — cos -^ 

/ 1 1 V a b 



(M x ) mai - , lV\o 8 + bv 

■*(? + *) 

(ilfv)»« = J- 7y ( ^2 + ^2 ) 

•"(sP + fti) 



(125) 



In the particular case of a square plate, a = b, and the foregoing 
formulas become 

uw = £^ (M x ) m „ = (M y ) m » = (1 + ^} qoa2 (126) 

We use Eqs. (106) and (107) to calculate the shearing forces and obtain 

~ q irx . iry 

Q x = 7— 1 v- cos — sin -£■ 

v» <> / (j7) 



-. g .irx vy 

Q y = — y-z sm — cos y 

^{h + hj 



SIMPLY SUPPORTED RECTANGULAR PLATES 



107 



To find the reactive forces at the supported edges of the plate we pro- 
ceed as was explained in Art. 22. For the edge x = a we find 



V x 



(«■ - ft. - - 



9o 



\a 2 ^ b 2 / 



ira 



2 \a* 



+ 



2 - v 



sin- 



(*) 



In the same manner, for the edge y 



V„ 



(* " SJ tl 



■b, 



■wb ( — 



i + i\ 



/l . 2 — A . the ,~ 



&v 



Hence the pressure distribution follows a sinusoidal law. The minus sign 
indicates that the reactions on the plate act upward. From symmetry 
it may be concluded that formulas (h) and (i) also represent pressure dis- 
tributions along the sides x = and y = 0, respectively. The resultant 
of distributed pressures is 



2g 






"1/1 , 2 - A P . Try 
a \a* + -F-J Jo Mn T 



d// 



+ 



2 



. 7T.r , 
sin — ax 
o a 



4g a?) 



is 



+ 



8g (l - ") 



■K-ab 



Observing that 



G 



lV 



4:q ab 



7T 



. 7T.T . 7T// 7 7 

</o sm — sin -V- ax az/ 
o a 6 



(J) 



(/<) 



it can be concluded that the sum of the distributed reactions is larger 
than the total load on the plate given by expression (k) . This result can 
be easily explained if we note that, proceeding as described in Art. 22, 
we obtain not only the distributed reactions 
but also reactions concentrated at the cor- 
ners of the plate. These concentrated re- 
actions are equal, from symmetry; and their 
magnitude, as may be seen from Fig. 51, is 



li — 2(M xy )x=a,y=b 



2?o(l - v) 



7T 



*ab 



\a 2 + by 



(I) 




rrTTT 



F' 



-I 

i 



Fig. 60 



R 

•J 



The positive sign indicates that the reactions act downward. Their sum 
is exactly equal to the second term in expression (j). The distributed 
and the concentrated reactions which act on the plate and keep the load, 
defined by Eq. (a), in equilibrium are shown graphically in Fig. 60. It 
may be seen that the corners of the plate have a tendency to rise up 



108 THEORY OF PLATES AND SHELLS 

under the action of the applied load and that the concentrated forces R 
must be applied to prevent this. 

The maximum bending stress is at the center of the plate. Assuming 
that a > b, we find that at the center M y > M x . Hence the maximum 
bending stress is 

f v _ 6(M y ) max 6 g0 (v_ 1\ 

The maximum shearing stress will be at the middle of the longer sides of 

the plate. Observing that the total transverse force V y = Q y -r-^ 

is distributed along the thickness of the plate according to the parabolic 
law and using Eq. (i), we obtain 

( ^ 3 g0 (I , 2- A 

out 1 4- lYV* a2 / 

If the sinusoidal load distribution is given by the equation 

g = go sin sin -r^ (raj 

where m and n are integer numbers, we can proceed as before, and we 
shall obtain for the deflection surface the following expression: 

w = — 7-^ — ^ sm sin -=-£• (127) 

a o 



*"><$*$) 



from which the expressions for bending and twisting moments can be 
readily obtained by differentiation. 

28. Navier Solution for Simply Supported Rectangular Plates. The 
solution of the preceding article can be used in calculating deflections 
produced in a simply supported rectangular plate by any kind of loading 
given by the equation 

q = f(x,y) ( a ) 

For this purpose we represent the function f(x,y) in the form of a double 
trigonometric series: 1 

00 00 

K*,y) = X X amn sin ^T sin ^T ^ 128 ^ 

m= 1 n= 1 

1 The first solution of the problem of bending of simply supported rectangular plates 
and the use for this purpose of double trigonometric series are due to Navier, who 



SIMPLY SUPPORTED RECTANGULAR PLATES 109 

To calculate any particular coefficient a m v of this series we multiply both 

sides of Eq. (128) by sin (n'iry /b) dy and integrate from to b. Observing 

that 

f b . mry . n'iry 7 ^ , , , 

/ sin —~ sin — r- 2 ay = when n 9^ n 
jo o b 

f b . mry . n'iry 7 b , , 

/ sin -r^ sin — ^ "2/ = 7> when n ~ n 
o b b J 2 



we find in this way 

oo 

/ /(>,?/) sin -y- d V = 2 £j amn ' Sm ~a~ 



GO 



m= 1 



Multiplying both sides of Eq. (6) by sin (m'lrx/a) dx and integrating from 
to a, we obtain 



£ /> 



N . mirx . n Tij 7 7 a& 

2/) sm sm — =— ax dy = -r- a m v 

a 6 4 



from which 



> „ . . mVx . n'iry , , , on * 

i i f( x ,y) sm sin ~ir^ ^ x «y (129) 

do lo Jo a b 



Performing the integration indicated in expression (129) for a given load 
distribution, i.e., for a given f(x,y), we find the coefficients of series (128) 
and represent in this way the given load as a sum of partial sinusoidal 
loadings. The deflection produced by each partial loading was discussed 
in the preceding article, and the total deflection will be obtained by sum- 
mation of such terms as are given by Eq. (127). Hence we find 

00 00 

1 V^ VA a mn ■ mirx . mry , _ m 

w = — -^ \ \ -r—^ ^ sin — sm -^- (130) 



771=1 71=1 \0 / 



Take the case of a load uniformly distributed over the entire surface 
of the plate as an example of the application of the general solution (130). 
Tn such a case 

ffay) = Qo 

where g is the intensity of the uniformly distributed load. From formula 
(129) we obtain 



ab Jo Jo 



a = ^ / / sin V™ sin VHM d x dv = -^- (c) 

- 3 ' ' a b ir 2 mn 



presented a paper on this subject to the French Academy in 1820. The abstract of the 
paper was published in Bull. soc. phil.-math., Paris, 1823. The manuscript is in the 
library of PEcole des Ponts et Chaussees. 



110 THEORY OF PLATES AND SHELLS 

where m and n are odd integers. If m or n or both of them are even 
numbers, a mn = 0. Substituting in Eq. (130), we find 

oo co . mirx . rtirii 

lO^o \^ \ A a, b , 101 . 

» = 3JTi > > T~* ^ (131) 






where m = 1, 3, 5, . . . and n = 1, 3, 5, . . . . 

In the ease of a uniform load we have a deflection surface symmetrical 
with respect to the axes x = a/2, g/ = 6/2; and quite naturally all terms 
with even numbers for m or n in series (131) vanish, since they are 
unsymmetrical with respect to the above-mentioned axes. The maxi- 
mum deflection of the plate is at its center and is found by substituting 
x = a/2, y = 6/2 in formula (131), giving 

.-IVV^^, 032) 






This is a rapidly converging series, and a satisfactory approximation is 
obtained by taking only the first term of the series, which, for example, 
in the case of a square plate gives 

Wmax = -^V/) = 0.00416 --p- 

or, by substituting expression (3) for D and assuming v = 0.3, 

goa 4 



= 0.0454 



Eh 1 



This result is about 2^- per cent in error (see Table 8). 

From expression (132) it may be seen that the deflections of two plates 
that have the same thickness and the same value of the ratio a/6 increase 
as the fourth power of the length of the sides. 

The expressions for bending and twisting moments can be obtained 
from the general solution (131) by using Eqs. (101) and (102). The 
series obtained in this way are not so rapidly convergent as series (131), 
and in the further discussion (see Art. 30) another form of solution will be 
given, more suitable for numerical calculations. Since the moments are 
expressed by the second derivatives of series (131), their maximum values, 
if we keep go and D the same, are proportional to the square of linear 
dimensions. Since the total load on the plate, equal to g a&, is also pro- 
portional to the square of the linear dimensions, we conclude that, for 
two plates of equal thickness and of the same value of the ratio a/6, the 



SIMPLY SUPPORTED RECTANGULAR PLATES 



111 



maximum bending moments and hence the maximum stresses are equal 
if the total loads on the two plates are equal. 1 

29. Further Applications of the Navier Solution. From the discussion 
in the preceding article it is seen that the deflection of a simply supported 
rectangular plate (Fig. 59) can always be represented in the form of a 
double trigonometric series (130), the coefficients a mn being given by 
Eq. (129). 

Let us apply this result in the case of a single load P uniformly dis- 
tributed over the area of the rectangle shown in Fig. 61. By virtue of 
Fq. (129) we have 

4P fZ + u/2 fv + v/2 



a n 



or 



a* 



abuv jt-u/2 
16P 



-v/2 



rrnrx . niry , 7 

sin sin -t— ax ay 

a b u 



. mir£ . utttj . rmru . rnrv 

sin sin -T- sin -= — sin -7^- 

ir 2 mnuv a b 2a 2b 



(a) 



If, in particular, £ = a/2, rj = 6/2, u = a, and 
n = 6, Eq. (a) yields the expression (c) obtained 
in Art. 28 for the uniformly loaded plate. 

Another case of practical interest is a single 
load concentrated at any given point x = £, y = 77 
of the plate. Using Eq. (a) and letting u and v 
tend to zero we arrive at the expression 



4P . nnr£ . nwn 

a mn = —r sin sin - 1 — 

ab a b 

and, by Eq. (130), at the deflection 



(&) 



1 


* 




<- u -> 


1 

V 

1 
1 

1 
Y 




1 


V/ 


V/ 


b 


l 
« 


Y/,Y/, 




— t— > 










1 


<— - 


-G - 




--> 





w = 



7T 



AP 






sin 



sin 



m=l n=l \a 2 ^ 6 2 / 



UTTT] 

~b~ 



sm 



Fig. 61 



rmrx . n-zr// 

sin -7— 

a b 



(133) 



The series converges rapidly, and we can obtain the deflection at any 
point of the plate with sufficient accuracy by taking only the first few 
terms of the series. Let us, for example, calculate the deflection at the 
middle when the load is applied at the middle as well. Then we have 

£ = x = a/2, 77 = y = 6/2, and the series (133) yields 






w u 




(c) 



1 This conclusion was established by Mariotte in the paper "Traite du mouvement 
<l<\s caiix," published in 1686. See Mariotte's scientific papers, new ed., vol. 2, p. 467, 
1710. 



112 THEORY OF PLATES AND SHELLS 

where m = 1, 3, 5, . . . and n = 1, 3, 5, ... . In the case of a square 
plate, expression (c) becomes 

oo oo 

Wm * x tt 4 D Lf Lf (w 2 + w 2 ) 2 

77i = 1 n — 1 

Taking the first four terms of the series we find that 

0.01121Pa 2 



"'max 



D 



which is about 3^- per cent less than the correct value (see Table 23, 
page 143). 

As for the series (128) representing the intensity of the concentrated 
load it is divergent at x = £, y = rj, and so also are the series expressing 
the bending moments and shearing forces at the point of application oi 
the load. 

Let us consider now the expression 



w = K(x,y,£,ij) = 



* oo . rrnr£ . mrn . rrnrx . nwv 

xr^\ \r^\ s i n sin -r 2 sin sin — - 

4 \ A \ A 



a o a 



m=l n = l \CL 2 b 2 J 



(134: 



which, by virtue of Eq. (133), represents the deflection due to a unit load 
P = 1 and for which the notation K(x,y,%,ri) is introduced for brevity. 

Regarding x and y as the variables, w = K(x,y,%,T)) is the equation oi 
the elastic surface of the plate submitted to a unit load at a fixed poinl 
x = £> V = *?• Now considering £ and 77 as variable, Eq. (134)' defines 
the influence surface for the deflection of the plate at a fixed point x, y 
the position of the traveling unit load being given by £ and 17. If, there- 
fore, some load of intensity f(%,rj) distributed over an area A is given, the 
corresponding deflection at any point of the plate may easily be obtained 
In fact, applying an elementary load /(£,*?) d% drj at x = £, y = 77 and 
using the principle of superposition, we arrive at the deflection 

W = // ffatiZb'V'M d * dr > < 135 ) 

A 

the double integral being extended over the loaded area and K(x,y,£,ri) 
being given by Eq. (134). / 

The function K(x,y,£,ii) is sometimes called Green's function of the plate. When 
given as by Eq. (134), this function is associated with the boundary conditions of the 
simply supported rectangular plate. Many properties of Green's function, however, 
are independent of those restrictions. An example is the property of symmetry, 



SIMPLY SUPPORTED RECTANGULAR PLATES 113 

expressed by the relation 

K(x,y,£, v ) = K(£, v ,x,y) 

which follows from the well-known reciprocal theorem of Maxwell 1 and is easy to 
verify in the particular case of the function (134). 

As the last example in the application of Navier's solution let us consider the case of 
as ingle load P uniformly distributed over the area of a circle with radius c and with 
center at x = £, y = 77. Introducing polar coordinates p, 6 with the origin at the 
center of the loaded area and replacing the elementary area dx dy in Eq. (129) by the 
area p dp dd, we have, by this latter equation, 



/:/: 



4 P C / * • rnTrO+pcos 6) . nn( v + p sin 6) 

dmn = — — : / / sin sin — p dp dd (a) 

ab ttc 2 



Provided that the circle p = c remains entirely inside the boundary of the plate the 
evaluation of the integral (d) gives the expression 2 

8P rrnri . nw V 

J i(y mnC) sin sin — ■ (e) 



~~ lit II j ^ »\ I III If-' / »J«" ►_!.»•» 

aocy mll a b 

in which y mn = tt \/(?n/a) 2 + (n/b) 2 and J\(y mn c) is the Bessel function of order one, 
with the argument y mn c. The required deflection now is obtainable by substitution 
of the expression (e) into Kq. (130). 

It is seen that the form of the Navier solution remains simple even in 
relatively complex cases of load distribution. On the other hand, the 
double series of this solution are not convenient for numerical computa- 
tion especially if higher derivatives of the function w are involved. So, 
another form of solution for the bending of the rectangular plate, more 
suitable for this purpose, will be discussed below. 

30. Alternate Solution for Simply Supported and Uniformly Loaded 
Rectangular Plates. In discussing problems of bending of rectangular 
plates that have two opposite edges simply supported, M. Levy 3 sug- 
gested taking the solution in the form of a series 



Y m sin — - (136) 

m— 1 



a 



where Y m is a function of y only. It is assumed that the sides x = and 
x = a (Fig. 62) are simply supported. Hence each term of series (136) 
satisfies the boundary conditions w = and d 2 w/dx 2 = on these two 
sides. It remains to determine Y m in such a form as to satisfy the bound- 

1 See, for instance, S. Timoshenko and D. H. Young, " Theory of Structures/' p. 
250, 1945. 

2 See S. Woinowsky-Krieger, Ingr.-Arch., vol. 3, p. 240, 1932. 

3 See Compi. rend., vol. 129, pp. 535-539, 1899. The solution was applied to several 
particular cases of bending of rectangular plates by E. Estanave, "Theses," Paris, 
1900; in this paper the transformation of the double series of the Navier solution to the 
simple series of M. Levy is shown. 



114 



THEORY OF PLATES AND SHELLS 



ary conditions on the sides // = + 6/2 and also the equation of the deflec- 
tion surface 



d 4 w 
dx 4 



+ 2 



d 4 W 
dx 2 dy 2 + 



d 4 w 
dy 4 



D 



(a) 



In applying this method to uniformly loaded and simply supported 

rectangular plates, a further simplification can be 
made by taking the solution of Eq. (a) in the form 1 




and letting 



W\ 



w = Wi + w 2 



(x 4 - 2ax 3 + a 3 x) 



24/) 



(&) 



(c) 



Fig. 62 



i.e., Wi represents the deflection of a uniformly 
loaded strip parallel to the x axis. It satisfies Eq. 
(a) and also the boundary conditions at the edges 
x = and x — a. 



The expression w 2 evidently has to satisfy the equation 





d 4 w 2 d 4 w 2 d 4 w 2 



(137) 



dx 4 ' ~ dx 2 dy 2 ' dy 4 

and must be chosen in such a manner as to make the sum (b) satisfy all 
boundary conditions of the plate. Taking w 2 in the form of the series 
(136) in which, from symmetry, to = 1, 3, 5, . . . and substituting it 
into Eq. (137), we obtain 



m= 1 



This equation can be satisfied for all values of x only if the function Y„ 
satisfies the equation 



1 



7IV 



mW~ v „ , 

6 5~~ ■» m T 



m 4 ir 4 







a a - a* 
The general integral of this equation can be taken in the form 2 



<<*) 



qa*/ 



, miry , D rrnry . , miry 
A m cosh — - + B m — - smh 



a 



a 



a 



+ C m sinh ™* + D m ™* cosh . 

a a a j 



miry \ 



(138) 



1 This form of solution was used by A. Nadai, Forschungsarb., nos. 170 and 171, 
Berlin, 1915; see also his book "Elastische Platten," Berlin, 1925. 

1 A somewhat different form for Y m , more convenient to satisfy some particular 
boundary conditions, has been suggested by P. F. Papkovitch, Priklad. Mat Mehh., 
vol. 5, 1941. 



SIMPLY SUPPORTED RECTANGULAR PLATES 115 

Observing that the deflection surface of the plate is symmetrical with 
respect to the x axis (Fig. 62), we keep in the expression (138) only even 
functions of y and let the integration constants C m = D ?n = 0. 

The deflection surface (b) is then represented by the following 
expression : 

w = 2T/) t* 4 " 2axZ + a%x ) 

oo 

+ 7T / \ A m cosh — - + B m — - smh — ^ 1 sm (e) 

which satisfies Eq. (a) and also the boundary conditions at the sides 
x = and x = a. It remains now to adjust the constants of integration 
.■!,„ and J5 m in such a manner as to satisfy the boundary conditions 

— o % = o (/) 

on the sides y = +b/2. We begin by developing expression (c) in a 
trigonometric series, which gives 1 

pr-^c (x 4 — 2a:c 3 + a z x) = -\jz / —i sm 

m = 1 

where m = 1, 3, 5, . . . . The deflection surface (e) will now be repre- 
sented in the form 



qa A V s / 4 . A , miry , „ ?n7r?y . , rmry\ . mmr , * 

w = V / I -T-i ; + A™ cosh — - -f Bm — L smh ) sm (g) 

D L*i \7r 5 m 5 a a a / a 



where m = 1, 3, 5, . . . . Substituting this expression in the boundary 
conditions (/) and using the notation 

rrnrb ,,. 

we obtain the following equations for determining the constants A m and 
4 



7r 5 m 5 



+ A M cosh a,„ + a m R m sinh a m = 



(A m + 2.7? m ) cosh <x m + ot m B m sinh a m = 
from which 

A = — 2 ( a ^ tan ^ a ™ + 2 ) R = 2 , ~ 

m ~"~ ir b m b cosh a,„ m 7r 5 ra 6 cosh a m 

1 See S. Timoshenko, "Strength of Materials," 3d ed., part II, p. 50, 1956. 



116 THEORY OF PLATES AND SHELLS 

Substituting these values of the constants in Eq. (g), we obtain the 
deflection surface of the plate, satisfying Eq. (a) and the boundary con- 
ditions, in the following form : 



oo 

= 4^ V iA _«,tanho, + 2 cosh - 
ir b D Z-/ m b \ 2 cosh a m 



a m 2y ^ 2a m y\ ._ rrnrx 



+ r r sinh — r-^ I sin (139) 

2 cosh a m b b J a 



from which the deflection at any point can be calculated by using tables 
of hyperbolic functions. 1 The maximum deflection is obtained at the 
middle of the plate (x = a/2, y = 0), where 

oo 

_ 4ga 4 V (-1) <«*-»/» / _ a m tanh a m + 2 \ ... 
™ m " ~~ ic b D Li m b \ 2 cosh a m J K3) 

m= 1,3,5,. .. 

Disregarding the second term in the parentheses, this series represents the 
deflection of the middle of a uniformly loaded strip. Hence we can 
represent expression (j) in the following form: 

00 

- _L 21* _ *2°-. V (-l) (m ~ 1)/2 «■» tanh "m + 2 

Wnu ~ 384 D x 8 I> Z/ w 6 2 cosh <* m *■ ; 

771= 1,3,5,. . . 

The series in this expression converges very rapidly, 2 and sufficient accu- 
racy is obtained by taking only the first term. Taking a square plate as 
an example, we know from Eq. (h) that 

tt Sir 

ai = 2 az = Y 



and Eq. (140) gives 

2£ _ Z^ (0.68562 - 0.00025 +•••)= 0.00406 ^r 
It is seen that the second term of the series in parentheses is negligible 



Wnmx = * qtf _ 4^ (0.68562 - 0.00025 +••■)= 0.00406 ^ 
384 



1 See, for example, "Tables of Circular and Hyperbolic Sines and Cosines/' 1939, 
and "Table of Circular and Hyperbolic Tangents and Cotangents,' ' 1943, Columbia 
University Press, New York; also British Association for the Advancement of Science, 
"Mathematical Tables," 3d ed., vol. 1, Cambridge University Press, 1951; finally, 
F. Losch, "Siebenstellige Tafeln der elementaren transzendenten Funktionen," 
Berlin, 1954. 

2 We assume that 6 ^ a, as in Fig. 62. 



SIMPLY SUPPORTED RECTANGULAR PLATES 117 

and that by taking only the first term the formula for deflection is 
obtained correct to three significant figures. 

Making use of the formula (140), we can represent the maximum 
deflection of a plate in the form 

w ma * = a ^- (141) 

where a is a numerical factor depending on the ratio b/a of the sides of 
the plate. Values of a are given in Table 8 (page 120). 

The bending moments M x and M y are calculated by means of expres- 
sion (e). Substituting the algebraic portion of this expression in Eqs. 
(101), we find that 

M , =2 *(a^ M ^ v ^iA (a) 

The substitution of the series of expression (e) in the same equations gives 

00 

M'J = (1 - v)qaW Y m 2 \ A m cosh ^ 

to= 1 



mwx 

(0 

rrnry 



, n /rrnry . , miry 2v , m7ry\l • m7r 

+ B m ( — - smh - -z cosh — - ) sin — 

\ a a 1 — v a J J a 

00 

M'J = -(1 - jOgaV 2 Y m 2 UL m cosh ^ 

m= 1 

+ B m ( — - sinh — J - + cosh — ~ J sin 

\ a a 1 — *> a /J a 

The total bending moments are obtained by summation of expressions 
(k) and (Z). Along the x axis the expression for the bending moments 
becomes 

00 

(IJh = ^^ - gaV 2 Y m 2 [2v£ ro - (1 - f)AJ sin ^ 

m— 1,3,5, . . . 

00 

(M,)^ = v qXi - a ~ X) ~ qaV Y m\2B m + (1 - v)^ m ] sin ^ 

m= 1,3,5, . . . 

Both series converge rapidly and the moments can readily be computed 
and represented in the form 

(M z ) y=0 = P'qa 2 (M y )y-o = & 9 a 2 Cm) 

The numerical values of the factors /?' and /3{ are given in Table 6. 



118 



THEORY OF PLATES AND SHELLS 



The bending moments acting along the middle line x = a/2 can be 
computed in a similar manner and represented in the form 



(M x ) x==a/2 = /3"ga 2 (M y ) x==a /2 = fc'qd 

Values of 0" and 0" are given in Table 7. 
The maximum values of these moments, 

(M,) max = (3qa 2 (M v ) m& , = foqa 2 



(n) 



(o) 



are at the center of the plate (x = a/2, y = 0), and the corresponding 
factors j8 and j8i are found in Table 8. The distribution of the moments 
in the particular case of a square plate is shown in Fig. 63. 

Table 6. Numerical Factors 0' and p\ for Bending Moments of Simply 
Supported Rectangular Plates under Uniform Pressure q 

v = 0.3, b >a 





M x = p'qa*, y = 


My = /3^a 2 , y == 


b/a 


x = 
0.1a 


x = 
0.2a 


x — 
0.3a 


x = 
0.4a 


x = 
0.5a 


x = 
0.1a 


x = 
0.2a 


x = 
0.3a 


x = 
0.4a 


x = 
0.5a 


1.0 
1.1 
1.2 
1.3 

1.4 

1.5 
1.6 
1.7 
1.8 
1.9 

2.0 
2.5 
3.0 
4.0 

00 


0.0209 
0.0234 
0.0256 
0.0277 
0.0297 

0.0314 
0.0330 
0.0344 
0.0357 
0.0368 

0.0378 
0.0413 
0.0431 
0.0445 
0.0450 


0.0343 
0.0389 
0.0432 
0.0472 
0.0509 

0.0544 
0.0572 
0.0599 
0.0623 
0.0644 

0.0663 
0.0729 
0.0763 
0.0791 
0.0800 


0.0424 
0.0486 
0.0545 
0.0599 
0.0649 

0.0695 
0.0736 
0.0773 
0.0806 
0.0835 

0.0861 
0.0952 
0.1000 
0.1038 
0.1050 


0.0466 
0.0541 
0.0607 
0.0671 
0.0730 

0.0783 
0.0831 
0.0874 
0.0913 
0.0948 

0.0978 
0.1085 
0.1142 
0.1185 
0.1200 


0.0479 
0.0554 
0.0627 
0.0694 
0.0755 

0.0812 
0.0862 
0.0908 
0:0948 
0.0985 

0.1017 
0.1129 
0.1189 
0.1235 
0.1250 


0.0168 
0.0172 
0.0174 
0.0175 
0.0175 

0.0173 
0.0171 
0.0169 
0.0167 
0.0165 

0.0162 
0.0152 
0.0145 
0.0138 
0.0135 


0.0303 
0.0311 
0.0315 
0.0316 
0.0315 

0.0312 
0.0309 
0.0306 
0.0301 
0.0297 

0.0292 
0.0272 
0.0258 
0.0246 
0.0240 


0.0400 
0.0412 
0.0417 
0.0419 
0.0418 

0.0415 
0.0411 
0.0405 
0.0399 
0.0393 

0.0387 
0.0359 
0.0340 
0.0322 
0.0315 


0.0459 
0.0475 
0.0480 
0.0482 
0.0481 

0.0478 
0.0472 
0.0466 
0.0459 
0.0451 

0.0444 
0.0412 
0.0390 
0.0369 
0.0360 


0.0479 
0.0493 
0.0501 
0.0503 
0.0502 

0.0498 
0.0492 
0.0486 
0.0479 
0.0471 

0.0464 
0.0430 
0.0406 
0.0384 
0.0375 



From Table 8 it is seen that, as the ratio b/a increases, the maximum 
deflection and the maximum moments of the plate rapidly approach the 
values calculated for a uniformly loaded strip or for a plate bent to a 
cylindrical surface obtained by making b/a = <*> . For b/a = 3 the dif- 
ference between the deflection of the strip and the plate is about 6£ per 
cent. For b/a = 5 this difference is less than £ per cent. The differ- 
ences between the maximum bending moments for the same ratios of 



SIMPLY SUPPORTED RECTANGULAR PLATES 



119 



Table 7. Numerical Factors p" and p" for Bending Moments of Simply 
Supported Rectangular Plates under Uniform Pressure q 

v = 0.3, b > a 





M x = 0"qa*, x = a/2 


My = p'l'qa*, x = a/2 


b/a 


y = 

0.4a 


V = 
0.3a 


V = 
0.2a 


y = 

0.1a 


y = o 


y = 
0.4a 


y = 
0.3a 


y = 

0.2a 


y = 

0.1a 


y = o 


1.0 
1.1 
1.2 
1.3 
1.4 

1.5 
1.6 

1.7 
1.8 
1.9 

2.0 
2.5 
3 
4 

00 


0.0168 
0.0197 
0.0225 
0.0252 
0.0275 

0.0302 
0.0324 
0.0348 
0.0371 
0.0392 

0.0413 
0.0505 
0.0586 
0.0723 
0.1250 


0.0303 
0.0353 
0.0401 
0.0447 
0.0491 

0.0532 
0.0571 
0.0607 
0.0641 
0.0673 

0.0703 
0.0828 
0.0923 
0.1054 
0.1250 


0.0400 
0.0465 
0.0526 
0.0585 
0.0639 

0.0690 
0.0737 
0.0780 
0.0819 
0.0854 

0.0887 
0.1012 
0.1092 
0.1180 
0.1250 


0.0459 
0.0532 
0.0600 
0.0667 
0.0727 

0.0781 
0.0832 
0.0877 
0.0917 
0.0953 

0.0986 
0.1102 
0.1168 
0.1224 
0.1250 


0.0479 
0.0554 
0.0627 
0.0694 
0.0755 

0.0812 
0.0862 
0.0908 
0.0948 
0.0985 

0.1017 
0.1129 
0.1189 
0.1235 
0.1250 


0.0209 
0.0225 
0.0239 
0.0252 
0.0203 

0.0275 
0.0288 
0.0295 
0.0304 
0.0314 

0.0322 
0.0360 
0.0389 
0.0426 
0.0375 


0.0343 
0.0363 
0.0379 
0.0391 
0.0402 

0.0410 
0.0417 
0.0423 
0.0428 
0.0433 

0.0436 
0.0446 
0.0447 
0.0436 
0.0375 


0.0424 
0.0442 
0.0454 
0.0462 
0.0468 

0.0470 
0.0471 
0.0470 
0.0469 
0.0467 

0.0464 
0.0447 
0.0431 
0.0406 
0.0375 


0.0466 
0.0481 
0.0490 
0.0494 
0.0495 

0.0493 
0.0489 
0.0484 
0.0478 
0.0472 

0.0465 
0.0435 
0.0413 
0.0389 
0.0375 


0.0479 
0.0493 
0.0501 
0.0503 
0.0502 

0.0498 
0.0492 
0.0486 
0.0479 
0.0471 

0.0464 
0.0430 
0.0406 
0.0384 
0.0375 



R = 0.065 qa^ 
i/ = 0.3 




Fig. 63 



120 



THEORY OF PLATES AND SHELLS 



b/a are 5 and •£ per cent, respectively. It may be concluded from this 
comparison that for b/a > 3 the calculations for a plate can be replaced 
by those for a strip without substantial error. 

Table 8. Numerical Factors «, /5, 7, 5, n for Uniformly Loaded and 

Simply Supported Rectangular Plates 

v = 0.3 





W max 


(M I ) ma x 


(M v ) ma x 


(Vz/niax 


(Vv/max 


( V x/nax 


( V i/jmax 


R 


b/a 




= pqa* 


= Piqa* 


= yqa 


= -yiqa 


= &qa 


= Siqa 


= nqa 1 




a 


P 


0i 


y 


71 


5 


Si 


n 


1.0 


0.00406 


0.0479 


0.0479 


0.338 


0.338 


0.420 


0.420 


0.065 


1.1 


0.00485 


0.0554 


0.0493 


0.360 


0.347 


0.440 


0.440 


0.070 


1.2 


0.00564 


0.0627 


0.0501 


0.380 


0.353 


0.455 


0.453 


0.074 


1.3 


0.00638 


0.0694 


0.0503 


0.397 


0.357 


0.468 


0.464 


0.079 


1.4 


0.00705 


0.0755 


0.0502 


0.411 


0.361 


0.478 


0.471 


0.083 


1.5 


0.00772 


0.0812 


0.0498 


0.424 


0.363 


0.486 


0.480 


0.085 


1.6 


0.00830 


0.0862 


0.0492 


0.435 


0.365 


0.491 


0.485 


0.086 


1.7 


0.00883 


0.0908 


0.0486 


0.444 


0.367 


0.496 


0.488 


0.088 


1.8 


0.00931 


0.0948 


0.0479 


0.452 


0.368 


0.499 


0.491 


0.090 


1.9 


0.00974 


0.0985 


0.0471 


0.459 


0.369 


0.502 


0.494 


0.091 


2.0 


0.01013 


0.1017 


0.0464 


0.465 


0.370 


0.503 


0.496 


0.092 


3.0 


0.01223 


0.1189 


0.0406 


0.493 


0.372 


0.505 


0.498 


0.093 


4.0 


0.01282 


0.1235 


0.0384 


0.498 


0.372 


0.502 


0.500 


0.094 


5.0 


0.01297 


0.1246 


0.0375 


0.500 


0.372 


0.501 


0.500 


0.095 


OO 


0.01302 


0.1250 


0.0375 


0.500 


0.372 


0.500 


0.500 


0.095 



Expression (e) can be used also for calculating shearing forces and 
reactions at the boundary. Forming the second derivatives of this 
expression, we find 



dx 2 ^ dy 2 2D ^ D Li a 

m=l 

Substituting this in Eqs. (106) and (107), we obtain 

Q. = Ht^M - 2, V t m'B. cosh =» -~ ™ X 

^ , X^ *r> • i_ miry . mirx 
Q y = — 2ir*qa > m z B m smh — - sin 



miry . mirx 
* sin 



a 



cos 



m=l 



SIMPLY SUPPORTED RECTANGULAR PLATES 121 

For the sides x = and y = — 6/2 we find 



00 

(Qx)x=o = 7j- ~ 27r 3 ga y m z B m cosh — 



oo 

_ qa _ 4<ja \^ 
~~2 1* Jif 



cosh^ 
a 

ra 2 cosh a m 



m= 1,3,5, , 



(Oy)v— 6/2 = 2-7r 3 ga / ra 3 jB m sinh a m sin 

771= 1 



7T 2 Z/ 



tanh a m . ra7rx 

= — sin 

ra 2 a 



m= 1,3,5, . .. 

These shearing forces have their numerical maximum value at the middle 
of the sides, where 

00 

(Q.)^-o = T - 7T 2/ m 2 cosh a m = 7(?a 

^=1,3.5,... (p) 



4ga y (-1)(«-d/2 

m= 1,3,5, 



(Qv)x=a/2, y =-b/2 = —j* / — o tanh a m = Yiga 



ra 2 



The numerical factors y and 71 are also given in Table 8. 

The reactive forces along the side x = are given by the expression 



00 
= (n dM *v \ = qa_4qa V^ 

\ dy Jx=o 2 ir 2 ^_^ 



I 



cosh^ 
a 

ra 2 cosh a m 

wi= 1,3,5, ... 



+ 2(1 - *)<?<* 



7r 2 Zy ra 2 cosh 2 « w 

m= 1,3,5, . . . 



(a m sinh a m cosh — cosh a m sinh — - | 
a a a J 

The maximum numerical value of this pressure is at the middle of the 
side (y = 0), at which point we find 



(T»)«-a,iM> = ? a 2 ~ ^ X 



1 



ra 2 cosh a w 

w= 1,3,5, ... 



, 2(1 - v) 



7T 

m = 1,3,5, 



X«m sinh a m \ f . 

m* cosh 2 a J = 3gQ (9) 



122 



THEORY OF PLATES AND SHELLS 



where 5 is a numerical factor depending on v and on the ratio b/a, which 
can readily be obtained by summing up the rapidly converging series 
that occur in expression (q). Numerical values of 5 and of Si, which 
corresponds to the middle of the sides parallel to the x axis, are given in 
Table 8. The distribution of the pressures (q) along the sides of a square 
plate is shown in Fig. 63. The portion of the pressures produced by the 



0.15 



0.10 



0.05 



































\0£_ — 




____ 










1§ 








































































& 












ft 





















































1.0 



1.5 



2.0 



2.5 u 3.0 
b 



3.5 



4.0 



Ratio 
Fig. 64 



twisting moments M xy is also shown. These latter pressures are bal- 
anced by reactive forces concentrated at the corners of the plate. The 
magnitude of these forces is given by the expression 

B - 2(M„)„,„ =W! = 2D(1 - ,) (^)_ 



_ 4(1 — v)qa 2 



i 



m 3 cosh a n 



m= l f 3,i5, 



y-6/2 

[(1 + a m tanh a m ) sinh a m 

— a m cosh a m ] = nqa 2 (r) 



The forces are directed downward and prevent the cornens of a plate 
from rising up during bending. The values of the coefficient n are given 
in the last column of Table 8. 



SIMPLY SUPPORTED RECTANGULAR PLATES 



123 



The values of the factors a, /?, p h 5 as functions of the ratio b/a are 
represented by the curves in Fig. 64. 

In the presence of the forces R, which act downward and are by no means small, 
anchorage must be provided at the corners of the plate if the plate is not solidly joined 
with the supporting beams. 

In order to determine the moments arising at the corner let us consider the equi- 
librium of the element abc of the plate next to its corner (Fig. 65) and let us introduce, 
for the same purpose, new coordinates 1, 2 at an angle of 45° to the coordinates x, y in 
Fig. 59. We can then readily verify that the bending moments acting at the sides ab 
and cb of the element are M i = — R/2 and M 2 = +R/2, respectively, and that the 
corresponding twisting moments are zero. In fact, using Eq. (39), we obtain for the 
side ac, that is, for the element of the edge, 
given by <x — —45°, the bending moment R 



Mr, = Mi cos 2 a + M 2 sin 2 a = 

in accordance with the boundary condi- 
tions of a simply supported plate. The 
magnitude of the twisting moment applied 
at the same edge element is obtained in like 
manner by means of Eq. (40). Putting 
a = —45° we have 

Mnt = - sin 2« (Mi — M 2 ) = — 



M 2 



17 


£^- 




\Mi 


2 


^J 


/4S° 




\45tW"' 


^^ 






a 


v 


b\ 


\ 







Fig. 65 



according to Eq. (r). Thus, the portion of the plate in the vicinity of the corner is 
bent to an anticlastic surface, the moments ±R/2 at the corner itself being of the 
same order of magnitude as the bending moments at the middle of the plate (see 
Table 8). 

The clamping effect of the corners of a simply supported plate is plainly illustrated 
by the distribution of the bending moments M x and M 2 of a square plate (Fig. 63). 
If the corners of the rectangular plate are not properly secured against lifting, the 
clamping becomes ineffective and the bending moments in the center portion of the 
plate increase accordingly. The values of (M z ) max and (M y ) ma x given in Table 8 
must then be multiplied by some factor k > 1. The approximate expression 1 



k = 



T V«* f + 



a 4 - f a 2 6 2 + 6 4 



may be used for that purpose. 

It should be noted that in the case of a polygonal plate with simply supported edges 
no single reactive forces arise at a corner point provided the angle between both 
adjacent sides of the plate is other than a right angle. 2 

Even in rectangular plates, however, no corner reactions are obtained if the trans- 
verse shear deformation is taken into account. In view of the strongly concentrated 

1 Recommended by the German Code for Reinforced Concrete (1943) and basod 
on a simplified theory of thin plates due to H. Marcus; see his book "Die vereinfachte 
Berechiuing biegsamer Flatten," 2d ed., Berlin, 1925. 

2 For a simple proof see, for example, H. Marcus, "Die Theorie elastischer Gewebe," 
2d ed., p. 46, Berlin, 1932. 



124 



THEORY OF PLATES AND SHELLS 



reactive forces this shear deformation obviously is no longer negligible, and the 
customary thin-plate theory disregarding it completely must be replaced by a more 
exact theory. The latter, which will be discussed in Art. 39, actually leads to. a dis- 
tribution of reactive pressures which include no forces concentrated at the corners of 
the plate (see Fig. 81). 

31. Simply Supported Rectangular Plates under Hydrostatic Pressure. 
Assume that a simply supported rectangular plate is loaded as shown in 
Fig. 66. Proceeding as in the case of a uniformly distributed load, we 
take the deflection of the plate in the form 1 



W = IPi + Wi 



in which 

_ go _ 
Wl ~ 360ZA a 



(£-io«. + 7..,)-^ % 



(-1) 



m+1 



m l 



sin 



rmrx 



a 



(a) 



(&) 



^lUl 



q X 



2 



TO 

t 

b 

5 

i 

i_ 



5° 



m= 1,2,3. ... 

represents the deflection of a strip under the tri- 
angular load. This expression satisfies the differ- 
ential equation 



d*w 



+ 2 



d 4 W 



+ 



d*W 



dx* ' " dx 2 dy* 
and the boundary conditions 
d 2 w 



= Q_ 

dy* D 



qpx 
aD 



(c) 



w = 



dx 2 



= f or x = and x = a 



Fig. 66 



The part ie 2 is taken in the form of a series 

oo 

Y m sin -£- (a) 



171=1 



where the functions Y m have the same form as in the preceding article, 
and m = 1, 2, 3, ... . Substituting expressions (6) and (d) into Eq. 
(a), we obtain 



w 



in — 1 



2(-l) 



tn+1 



7r 6 m 6 



+ A m cosh ™* + B 



^y sinh m*v\ sin ™*h 

a a J a 



(e) 
where the constants A m and B m are to be determined from the conditions 



w = 



^ = 

ay 2 



f or y = +2 



' This problem was discussed by E. Estanave, op. cit. The numerical tables of 
deflections and moments were calculated by B. G. Galerkin, Bull. Polytech. Inst., St. 
Petersburg, vols. 26 and 27, 1918. 



SIMPLY SUPPORTED RECTANGULAR PLATES 125 

From these conditions we find 
2(-l)«+i 



7r 5 m 5 



+ A m cosh a m + B m a m sinh a m = 



(2B m + i m ) cosh <x m + 5 m a m sinh a m = 
In these equations we use, as before, the notation 

rrnrb 



OCm = 



2a 

Solving them, we find 

A (2 + a m tanhQ(-l)"+ 1 R = (-!)-+* (/) 

m 7r 5 m 5 cosh a m m 7r 5 ra 5 cosh a m 

The deflection of the plate along the x axis is 

00 



. mwx 
m I sin 



a 

m — l 



For a square plate a = 6, and we find 



£(o. 



(u,) = 2°£ ^0.002055 sin — - 0.000177 sin — 



+ 0.000025 sin ~- - • • • J (g) 



a 



The deflection at the center of the plate is 



O)*~a/2,^0 = 0.00203 2g^ (h) 

which is one-half the deflection of a uniformly loaded plate (see page 116) 
as it should be. By equating the derivative of expression (g) to zero, we 
find that the maximum deflection is at the point x = 0.557a. This maxi- 
mum deflection, which is 0.00206 q a A /D, differs only very little from the 
deflection at the middle as given by formula (h). The point of maximum 
deflection approaches the center of the plate as the ratio b/a increases. 
For b/a = <*>, as for a strip [see expression (&)], the maximum deflection 
is at the point x = 0.5193a. When b/a < 1, the point of maximum 
deflection moves away from the center of the plate as the ratio b/a 
decreases. The deflections at several points along the x axis (Fig. 66) 
are given in Table 9. It is seen that, as the ratio b/a increases, the 
deflections approach the values calculated for a strip. For b/a = 4 the 
differences in these values are about 1% per cent. We can always calcu- 
late the deflection of a plate for which b/a > 4 with satisfactory accu- 
racy by using formula (6) for the deflection of a strip under triangular 
load. The bending moments M x and M y are found by substituting 



126 



THEORY OF PLATES AND SHELLS 



Table 9. Numerical Factor a for Deflections of a Simply Supported 
Rectangular Plate under Hydrostatic Pressure q = qox/a 

b > a 

w = aq a 4 /D, y = 



b/a 


x = 0.25a 


x = 0.50a 


x = 0.60a 


x = 0.75a 


1 


0.00131 


0.00203 


0.00201 


0.00162 


1.1 


0.00158 


0.00243 


0.00242 


0.00192 


1.2 


0.00186 


0.00282 


0.00279 


0.00221 


1.3 


0.00212 


0.00319 


0.00315 


0.00248 


1.4 


0.00235 


0.00353 


0.00348 


0.00273 


1.5 


0.00257 


0.00386 


0.00379 


0.00296 


1.6 


0.00277 


0.00415 


0.00407 


0.00317 


1.7 


0.00296 


0.00441 


0.00432 


0.00335 


1.8 


0.00313 


0.00465 


0.00455 


0.00353 


1.9 


0.00328 


0.00487 


0.00475 


0.00368 


2.0 


0.00342 


0.00506 


0.00494 


0.00382 


3.0 


0.00416 


0.00612 


0.00592 


0.00456 


4.0 


0.00437 


0.00641 


0.00622 


0.03477 


5.0 


0.00441 


0.00648 


0.00629 


0.00483 


00 


0.00443 


0.00651 


0.00632 


0.00484 



expression (e) for deflections in Eqs. (101). Along the x axis (y = 0) 
the expression for M x becomes 



X2( — l) m+1 . m-KX 
T . m » sm 



a 



rmrx 



+ qoaV ) m 2 [(l - v)A m - 2vB m \ sin ^ (i) 



I 

m= 1 

The first sum on the right-hand side of this expression represents the 
bending moment for a strip under the action of a triangular load and is 
equal to (g /6)(ax - x*/a). Using expressions (/) for the constants A m 
and B m in the second sum, we obtain 



iM\ go(a 2 s ~ * 3 ) 



6a 



_goa 2 V J 

m = l 



' [2 + (1 - v)a m tanh a m ] sin —^ 0) 



3 cosh a T 



a 



The series thus obtained converges rapidly, and a sufficiently accurate 
value of M x can be realized by taking only the first few terms. In this 



SIMPLY SUPPORTED RECTANGULAR PLATES 



127 



Table 10. Numerical Factors /3 and Pi for Bending Moments of Simply 
Supported Rectangular Plates under Hydrostatic Pressure q = qox/a 

v = 0.3, b > a 





M x = /3a 2 g 0) V = 


My = fiia 1 


go, y = 1 





b/a 


x = 


x = 


x = 


x = 


x = 


x = 


x = 


x = 




0.25a 


0.50a 


0.60a 


0.75a 


0.25a 


0.50a 


0.60a 


0.75a 


1.0 


0.0132 


0.0239 


0.0264 


0.0259 


0.0149 


0.0239 


0.0245 


0.0207 


1.1 


0.0156 


0.0276 


0.0302 


0.0289 


0.0155 


0.0247 


0.0251 


0.0211 


1.2 


0.0179 


0.0313 


0.0338 


0.0318 


0.0158 


0.0250 


0.0254 


0.0213 


1.3 


0.0200 


0.0346 


0.0371 


0.0344 


0.0160 


0.0252 


0.0255 


0.0213 


1.4 


0.0221 


0.0376 


0.0402 


0.0367 


0.0160 


0.0253 


0.0254 


0.0212 


1.5 


0.0239 


0.0406 


0.0429 


0.0388 


0.0159 


0.0249 


0.0252 


0.0210 


1.6 


0.0256 


0.0431 


0.0454 


0.0407 


0.0158 


0.0246 


0.0249 


0.0207 


1.7 


0.0272 


0.0454 


0.0476 


0.0424 


0.0155 


0.0243 


0.0246 


0.0205 


1.8 


0.0286 


0.0474 


0.0496 


0.0439 


0.0153 


0.0239 


0.0242 


0.0202 


1.9 


0.0298 


0.0492 


0.0513 


0.0452 


0.0150 


0.0235 


0.0238 


0.0199 


2.0 


0.0309 


0.0508 


0.0529 


0.0463 


0.0148 


0.0232 


0.0234 


0.0197 


3.0 


0.0369 


0.0594 


0.0611 


0.0525 


0.0128 


0.0202 


0.0207 


0.0176 


4.0 


0.0385 


0.0617 


0.0632 


0.0541 


0.0120 


0.0192 


0.0196 


0.0168 


5.0 


0.0389 


0.0623 


0.0638 


0.0546 


0.0118 


0.0187 


0.0193 


0.0166 


00 


0.0391 


0.0625 


0.0640 


0.0547 


0.0117 


0.0187 


0.0192 


0.0165 



way the bending moment at any point of the x axis can be represented 
by the equation 

(Af«)„-o = /fyoa 2 (fc) 

where /3 is a numerical factor depending on the abscissa x of the point. 
In a similar manner we get 

(Af y )„«o = /3i(7oa 2 (0 

The numerical values of the factors and ft in formulas (fc) and (I) are 
given in Table 10. It is seen that for & ^ 4a the moments are very close 
to the values of the moments in a strip under a triangular load. 

Equations (106) and (107) are used to calculate shearing forces. From 
the first of these equations, by using expression (j), we obtain for points 
on the x axis 



(Q.)^o = - 



d^ /dhv d 2 w \ 
dx\dx 2 + dy 2 )y 

qo(a 2 - 3x 2 ) 



f y=0 



Qa 



00 

_ 2qoa Y 

7T 2 Li 



(-D 



tn+1 



m 2 cosh a* 



cos 



rmrx 
a 



128 THEORY OF PLATES AND SHELLS 

The general expressions for shearing forces Q x and Q v are 

miry 



Q.- 



^ ^ (-l) m+1 cosh 
q (a* - 3s 2 ) 2q a X ^ a 



6a 



I 

ro=l 



& 



00 

_ _ 2q a V^ 

= *' Li 



(-l)" +1 sinh 



ra 2 cosh a„ 



m 2 cosh a„ 



m7ry 

a . rmrx 

- sin 



cos (m) 

a 



a 



(n) 



The magnitude of the vertical reactions V x and V y along the boundary 
is obtained by combining the shearing forces with the derivatives of the 
twisting moments. Along the sides x = and x = a these reactions can 
be represented in the form 






(o) 



Table 11. Numerical Factors 5 and 6i for Reactions op Simply 
Supported Rectangular Plates under Hydrostatic Pressure q = qax/a 

v = 0.3, b > a 





Reactions Sq^a 


Reactions 5igo6 






x = 


■ 


x = 


■■ a 




V = : 


±6/2 




b/a 




























/ 


















y — 




y = 


x = 


x = 


x = 


x = 




y 


= 


0.256 


y = 


0.256 


0.25a 


0.50a 


0.60a 


0.75a 


1.0 


0. 


126 


0.098 


0.294 


0.256 


0.115 


0.210 


0.234 


0.239 


1.1 


0. 


136 


0.107 


0.304 


0.267 


0.110 


0.199 


0.221 


0.224 


1.2 


0. 


144 


0.114 


0.312 


0.276 


0.105 


0.189 


0.208 


0.209 


1.3 


0. 


150 


0.121 


0.318 


0.284 


0.100 


0.178 


0.196 


0.196 


1.4 


0. 


155 


0.126 


0.323 


0.292 


0.095 


0.169 


0.185 


0.184 


1.5 


0. 


159 


0.132 


0.327 


0.297 


0.090 


0.160 


0.175 


0.174 


1.6 


0. 


162 


0.136 


0.330 


0.302 


0.086 


0.151 


0.166 


0.164 


1.7 


0. 


164 


0.140 


0.332 


0.306 


0.082 


0.144 


0.157 


0.155 


1.8 





166 


0.143 


0.333 


0.310 


0.078 


0.136 


0.149 


0.147 


1.9 





167 


0.146 


0.334 


0.313 


0.074 


0.130 


0.142 


0.140 


2.0 





.168 


0.149 


0.335 


0.316 


0.071 


0.124 


0.135 


0.134 


3.0 





169 


0.163 


0.336 


0.331 


0.048 


0.083 


0.091 


0.089 


4.0 





.168 


0.167 


0.334 


0.334 


0.036 


0.063 


0.068 


0.067 


5.0 





.167 


0.167 


0.334 


0.335 


0.029 


0.050 


0.055 


0.054 


00 





.167 


0.167 


0.333 


0.333 











SIMPLY SUPPORTED RECTANGULAR PLATES 



129 



and along the sides y = + b/2 in the form 



\ OX /y=±b/2 



(P) 



in which 5 and 5 X are numerical factors depending on the ratio b/a and 
on the coordinates of the points taken on the boundary. Several values 
of these factors are given in Table 11. 

The magnitude of concentrated forces that must be applied to prevent 
the corners of the plate rising up during bending can be found from the 
values of the twisting moments M xy at the corners. Since the load is not 
symmetrical, the reactions R\ at x = and y = ±6/2 are different from 
the reactions R 2 at x = a and y = ±6/2. These reactions can be repre- 
sented in the following form : 

R\ = niqoab R 2 = n 2 qoab (q) 

The values of the numerical factors n\ and n 2 are given in Table 12. 



Table 12. Numerical Factors n\ and n 2 in Eqs. (7) for Reactive Forces 

R\ AND 72 2 AT THE CORNERS OF SlMPLY SUPPORTED RECTANGULAR PLATES 

under Hydrostatic Pressure q = qox/a 
v = 0.3, b > a 



b/a 



712 



1.0 



1.1 



0.026 0.026 
0.039 0.038 



1.2 



0.026 
0.037 



1.3 



1.4 



1.5 



1.6 



1.7 



0.026 0.025 0.024 0.023 0.022 
0.036,0.035 0.033 0.032 0.030 



1.8 



0.021 
0.029 



1.9 



0.021 
0.028 



2.0 



0.020 
0.026 



3.0 



0.014 
0.018 



4.0 



5.0 



0.010 0.008 
0.014 0.011 



Since a uniform load q Q is obtained by superposing the two triangular 
loads q = qox/a and qo(a — x)/a, it can be concluded that for correspond- 
ing values of b/a the sum n\ + n 2 of the factors given in Table 12 multi- 
plied by b/a must equal the corresponding value of n, the last column in 
Table 8. 

If the relative dimensions of the plate are such that a in Fig. 66 is 
greater than b, then more rapidly converging series will be obtained by 
representing W\ and w 2 by the following expressions: 



wi = ~ gg^ ( lQ y 4 - 246 v + 564 ) 



w 2 



00 

-I 

Wl= 1 



Xtm-l COS 



(2m — l)iry 



(r) 



(*) 



The first of these expressions is the deflection of a narrow strip parallel to 
the y axis, supported at y = + 6/2 and carrying a uniformly distributed 



130 



THEORY OF PLATES AND SHELLS 



Table 13. Numerical Factors a for Deflections of Simply Supported 
Rectangular Plates under Hydrostatic Pressure q = qox/a 

b < a 
w = <xqob 4 /D, y = 



a/b 


x = 0.25a 


x = 0.50a 


x = 0.60a 


x = 0.75a 


00 


0.00325 


0.00651 


0.00781 


0.00976 


5 


0.00325 


0.00648 


0.00778 


0.00965 


4 


0.00325 


0.00641 


0.00751 


0.00832 


3 


0.00321 


0.00630 


0.00692 


0.00707 


2 


0.00288 


0.00506 


0.00542 


0.00492 


1.9 


0.00281 


0.00487 


0.00518 


0.00465 


1.8 


0.00270 


0.00465 


0.00491 


0.00434 


1.7 


0.00261 


0.00441 


0.00463 


0.00404 


1.6 


0.00249 


0.00415 


0.00432 


0.00372 


1.5 


0.00234 


0.00386 


0.00399 


0.00339 


1.4 


0.00218 


0.00353 


0.00363 


0.00304 


1.3 


0.00199 


0.00319 


0.00325 


0.00269 


1.2 


0.00179 


00282 


0.00286 


0.00234 


1.1 


0.00153 


0.00243 


0.00245 


0.00199 


1.0 


0.00131 


0.00202 


0.00201 


0.00162 



.till 




* 


\\\ii 


qo 


z 


(a) 


* 



load of intensity qox/a. This expression satisfies the differential equa- 
tion (c) and also the boundary conditions w = and d*w/dy* = at 

y = + 6/2. Expression (s) represents an in- 
finite series each term of which also satisfies 
the conditions at the edges y = + 6/2. The 
functions X 2m -i of x are chosen in such a 
manner that each of them satisfies the homo- 
geneous equation (137) of the preceding arti- 
cle (see page 114) and so that expression (a) 
satisfies the boundary conditions at the edges 
x = and x = a. Since the method of de- 
termining the functions X^m-i is similar to 
that already used in determining the func- 
tions F m , we shall limit ourselves to giving 
only the final numerical results, which are 
represented by Tables 13, 14, 15, and 16. 
The notation in these tables is the same as 
in the foregoing tables for the hydrostatic pressure. 

32. Simply Supported Rectangular Plate under a Load in the Form of 
a Triangular Prism. Assume that the intensity of the load is represented 




SIMPLY SUPPORTED RECTANGULAR PLATES 



131 



Table 14. Numerical Factors and 0i for Bending Moments in Simply 
Supported Rectangular Plates under Hydrostatic Pressure q = qa/a 

v = 0.3, b < a 





M, = /3& 2 g , y = 


M u = /3i6 2 g , y = 


a/b 


x = 


x = 


x = 


x = 


x = 


x = 


x = 


x = 




0.25a 


0.50a 


0.60a 


0.75a 


0.25a 


0.50a 


0.60a 


0.75a 


00 


0.0094 


0.0187 


0.0225 


0.0281 


0.0312 


0.0625 


0.0750 


0.0937 


5.0 


0.0094 


0.0187 


0.0230 


0.0309 


0.0312 


0.0623 


0.0742 


0.0877 


4.0 


0.0094 


0.0192 


0.0237 


0.0326 


0.0312 


0.0617 


0.0727 


0.0820 


3.0 


0.0096 


0.0202 


0.0256 


0.0345 


0.0309 


0.0594 


0.0678 


0.0715 


2.0 


0.0108 


0.0232 


0.0285 


0.0348 


0.0284 


0.0508 


0.0554 


0.0523 


1.9 


0.0111 


0.0235 


0.0288 


0.0345 


0.0278 


0.0492 


0.0533 


0.0498 


1.8 


0.0115 


0.0239 


0.0291 


0.0341 


0.0269 


0.0474 


0.0509 


0.0470 


1.7 


0.0117 


0.0243 


0.0293 


0.0337 


0.0261 


0.0454 


0.0485 


0.0442 


1.6 


0.0120 


0.0246 


0.0294 


0.0331 


0.0251 


0.0431 


0.0457 


0.0412 


1.5 


0.0123 


0.0249 


0.0294 


0.0324 


0.0239 


0.0406 


0.0428 


0.0381 


1.4 


0.0126 


0.0253 


0.0292 


0.0315 


0.0225 


0.0376 


0.0396 


0.0348 


1.3 


0.0129 


0.0252 


0.0290 


0.0304 


0.0209 


0.0346 


0.0360 


0.0314 


1.2 


0.0131 


0.0250 


0.0284 


0.0291 


0.0192 


0.0313 


0.0323 


0.0279 


1.1 


0.0134 


0.0247 


0.0276 


0.0276 


0.0169 


0.0276 


0.0285 


0.0245 


1.0 


0.0132 


0.0239 


0.0264 


0.0259 


0.0149 


0.0239 


0.0245 


0.0207 



by an isosceles triangle as shown in Fig. 67a. 
again be represented in the form 



The deflection surface can 



(a) 



in which w x represents the deflection of a simply supported strip parallel 
to the x axis, and w 2 has the same form as in the preceding article [Eq. (d)]. 
To represent the deflection w x in the form of a trigonometric series we 
observe that the deflection produced by a concentrated force P applied 
at a distance £ from the left end of a strip is 1 



2Pa 3 V J_ . 

Dtt 4 Li m* S1 

m — 1 



mir£ . rmrx 
sm sin 



a 



a 



(b) 



Substituting q&% for P and using q = 2q Z/a for £ < a/2 and 
q = 2q (a - £)/a for £ > a/2, the deflection of the strip by an ele- 
mental load is obtained. The deflection produced by the total load on 

i See Timoshenko, "Strength of Materials," part II, 3d ed., p. 49, 1956. 



132 



THEORY OF PLATES AND SHELLS 



Table 15. Numerical Factors 5 and 5i for Reactions in Simply Supported 
Rectangular Plates under Hydrostatic Pressure q = qox/a 

v = 0.3, b <a 





Reactions 5g a 


Reactions $iq<jb 


a/b 


X ' 


= 


x ■■ 


= a 




y = 


+ 6/2 






y = 


y =6/4 


y - 


y =6/4 


x = 
0.25a 


X = 

0.50a 


x = 
0.60a 


x = 
0.75a 


00 










0.125 


0.250 


0.300 


0.375 


5.0 


0.008 


0.006 


0.092 


0.076 


0.125 


0.250 


0.301 


0.379 


4.0 


0.013 


0.010 


0.112 


0.093 


0.125 


0.251 


0.301 


0.377 


3.0 


0.023 


0.018 


0.143 


0.119 


0.125 


0.252 


0.304 


0.368 


2.0 


0.050 


0.038 


0.197 


0.166 


0.127 


0.251 


0.296 


0.337 


1.9 


0.055 


0.041 


0.205 


0.172 


0.127 


0.251 


0.294 


0.331 


1.8 


0.060 


0.045 


0.213 


0.179 


0.128 


0.249 


0.291 . 


0.325 


1.7 


0.066 


0.050 


0.221 


0.187 


0.127 


0.248 


0.288 


0.318 


1.6 


0.073 


0.055 


0.230 


0.195 


0.127 


0.245 


0.284 


0.311 


1.5 


0.080 


0.060 


0.240 


0.204 


0.127 


0.243 


0.279 


0.302 


1.4 


0.088 


0.067 


0.250 


0.213 


0.126 


0.239 


0.273 


0.292 


'1.3 


0.097 


0.074 


0.260 


0.223 


0.124 


0.234 


0.266 


0.281 


1.2 


0.106 


0.081 


0.271 


0.233 


0.122 


0.227 


0.257 


0.269 


1.1 


0.116 


0.090 


0.282 


0.244 


0.120 


0.220 


0.247 


0.255 


1.0 


0.126 


0.098 


0.294 


0.256 


0.115 


0.210 


0.234 


0.239 



the strip is now obtained by integration in the following form: 

4g a 2 V ! • mirX T f° /2 y • mw £ jy . f° / >\ • m ^ jJ\ 

Wl = iw 2, m* sm ir [Jo * s ™-ir d z + J a/2 ( a - «*-£-'*] 



m=l 



— 8go a 



00 

m = 1,3,5, 



(_1) <«-!>/* . mirx 



m" 



sin 



a 



(c) 



Substituting this in Eq. (a) and using Eq. (d) of the preceding article, 
we obtain 



w 



00 

= goa 4 V f 8( — l)^- 1 )/ 2 

D Li L **™>* 



m= 1,3,5, ... 



+ A m cosh — - + 2? m — ^ smh — - sin 

a a a J a 



(<*) 



This expression satisfies Eq. (103) and also the boundary conditions at 
the edges x = and x = a. The constants ^4 m and Z? m can be found 



SIMPLY SUPPORTED RECTANGULAR PLATES 



133 



from the conditions along the edges y = ± 6/2, which are the same as in 
the preceding article and which give 



8(-l) 



(m-l)/2 



7T 6 ?n 6 



+ A m cosh a m + B m a m sinh a m = 



(2B m + A m ) COSh « m + BmOCm SlTlh <X m = 

where, as before, we use the notation 

rnirb 



« 



a m = 



2a 



Solving Eqs. (e), we find 

. __ 4(2 + a m tanha m )(-l)<"-D/2 

^m — 
#m = 



7r 6 m 6 cosh a v 

4(_l)(m-l)/2 



(/) 



7r 6 m 6 cosh a m 

To obtain the deflection of the plate along the x axis we put y = in 

Table 16. Numerical Factors n\ and n 2 in Eqs. (g) (Art. 31) for Reactive 

Forces R\ and R 2 at the Corners of Simply Supported Rectangular 

Plates under Hydrostatic Pressure g = qox/a 

v = 0.3, b < a 



a/b 


5 


4 


3 


2 


1.9 


1.8 


1.7 


1.6 


1.5 


1.4 


1.3 


1.2 


1.1 


1.0 


ni 


0.002 
0.017 


0.004 
0.020 


0.006 
0.025 


0.013 
0.033 


0.014 
0.034 


0.016 
0.035 


0.017 
0.036 


0.018 
0.037 


0.020 
0.037 


0.021 
0.038 


0.023 
0.039 


0.024 
0.039 


0.025 
0.039 


0.026 
0.039 



expression (d). Then 



(w)y= 



goat V [ 8(-l)("-*>/* 1 

D L( L 7r 6 m 6 + Am \ 



sin 



rrnrx 



a 



m = 1,3,5, . .. 

The maximum deflection is at the center of the plate, where 



W D 



m = l,3,5, , 

It can be represented in the form 



-# t fe+^-^-j 



Wmax = « 



_ ?oa 4 



D 



in which a is a numerical factor depending on the magnitude of the ratio 
b/a. Several values of this factor are given in Table 17. l 
1 The tables are taken from the paper by Galerkin, loc. cit. 



134 



THEORY OF PLATES AND SHELLS 



Table 17. Numerical Factors a, 0, y, 5, n for Simply Supported 
Rectangular Plates under a Load in Form of a Triangular Prism 

v = 0.3, b > a 





Wmax 


















qoa 4 


W" x)ma.x 


(iVZ yjmax 


\ijx)mnx 


\%£y)m&x 


\ V x J max 


\ y y/inax 


R 


b/a 


= a ~D 


= 0goa 2 


= /Sigoo 2 


= 7^00 


= yiqob 


= dqoa 


= 5igo& 


— nqoab 




a 





0. 


y 


71 


s 


Si 


n 


1.0 


0.00263 


0.0340 


0.0317 


0.199 


0.315 


0.147 


0.250 


0.038 


1.1 


0.00314 


0.0390 


0.0326 


0.212 


0.297 


0.161 


0.232 


0.038 


1.2 


0.00364 


0.0436 


0.0330 


0.222 


0.280 


0.173 


0.216 


0.037 


1.3 


0.00411 


0.0479 


0.0332 


0.230 


0.265 


0.184 


0.202 


0.036 


1.4 


0.00455 


0.0518 


0.0331 


0.236 


0.250 


0.193 


0.189 


0.035 


1.5 


0.00496 


0.0554 


0.0329 


0.241 


0.236 


0.202 


0.178 


0.034 


1.6 


0.00533 


0.0586 


0.0325 


0.246 


0.224 


0.208 


0.168 


0.033 


1.7 


0.00567 


0.0615 


0.0321 


0.247 


0.212 


0.214 


0.158 


0.031 


1.8 


0.00597 


0.0641 


0.0316 


0.249 


0.201 


0.220 


0.150 


0.030 


1.9 


0.00625 


0.0664 


0.0311 


0.251 


0.191 


0.224 


0.142 


0.029 


2.0 


0.00649 


0.0685 


0.0306 


0.252 


0.183 


0.228 


0.135 


0.028 


3.0 


0.00783 


0.0794 


0.0270 


0.253 


0.122 


0.245 


0.090 


0.019 


00 


0.00833 


0.0833 


0.0250 


0.250 




0.250 







Using expression (d) and proceeding as in the preceding article, we can 
readily obtain the expressions for bending moments M x and M y . The 
maximum values of these moments in this case are evidently at the center 

of the plate and can be represented in the fol- 
lowing form: 



t-L 



UlUv viulluj t 



2 2 

Fig. 68 



(M x ) max = Pqoa 2 (M y ) m * x = p x q a 2 



The values of the numerical factors j8 and /3i are 
also given in Table 17. This table also gives 
numerical factors y, 71, 8, 61, and n for calculating (1) shearing forces 
(Qz)m*x = yqoa, (Q y )m*x = 7i?o& at the middle of the sides x = and 
y = —6/2 of the plate, (2) reactive forces 



* = (q x - 



dM 



U / max 
J max 



8qod 



8iqob 



at the same points, and (3) concentrated reactions R = nq^ab at the 
corners of the plate which are acting downward and prevent the corners 
of the plate from rising. All these values are given for b > a. When 
b < a, a better convergency can be obtained by taking the portion Wi 



SIMPLY "SUPPORTED RECTANGULAR PLATES 



135 



Table 18. Numerical Factors a, 0, y, 5, n for Simply Supported 
Rectangular Plates under a Load in Form of a Triangular Prism 

v = 0.3, b < a 



1 
1 

1 


1*) max 


(M x ) mxx 


(M y ) max 


VVz/max 


(^Jy)max 


( v xjniax 


( V j/jmax 


R 


a/b 


= /3go& 2 


= Piqob* 


= 79 oa 


= 7i<Zob 


= Sqod 


= Sigob 


= nqoab 




a 





Pi 


y 


71 


8 


5i 


n 


00 


0.01302 


0.0375 


0.1250 




0.500 




0.500 




3.0 


0.00868 


0.0387 


0.0922 


0.045 


0.442 


0.027 


0.410 


0.010 


2.0 


0.00686 


0.0392 


0.0707 


0.091 


0.412 


0.057 


0.365 


0.023 


1.9 


0.00656 


0.0392 


0.0681 


0.098 


0.407 


0.062 


0.358 


0.024 


1.8 


0.00624 


0.0391 


0.0651 


0.106 


0.402 


0.098 


0.350 


0.026 


1.7 


0.00588 


0.0390 


0.0609 


0.115 


0.396 


0.074 


0.342 


0.028 


1.6 


0.00549 


0.0388 


0.0585 


0.124 


0.389 


0.081 


0.332 


0.029 


1.5 


0.00508 


0.0386 


0.0548 


0.135 


0.381 


0.090 


0.322 


0.031 


1.4 


0.00464 


0.0382 


0.0508 


0.146 


0.371 


0.099 


0.311 


0.033 


1.3 


0.00418 


0.0376 


0.0464 


0.158 


0.360 


0.109 


0.298 


0.035 


1.2 


0.00367 


0.0368 


0.0418 


0.171 


0.347 


0.120 


0.284 


0.036 


1.1 


0.00316 


0.0356 


0.0369 


0.185 


0.332 


0.133 


0.268 


0.037 


1.0 


0.00263 


0.0340 


0.0317 


0.199 


0.315 


0.147 


0.250 


0.038 



-Q|(\J 



4- 



.o|c\J 



of the deflection of the plate in the form of the deflection of a strip 
parallel to the y direction. We omit the derivations and give only the 
numerical results assembled in Table 18. 

Combining the load shown in Fig. 67a with 
the uniform load of intensity g , the load 
shown in Fig. 68 is obtained. Information 
regarding deflections and stresses in this lat- 
ter case can be obtained by combining the 
data of Table 8 with those of Table 17 or 18. 

33. Partially Loaded Simply Supported 
Rectangular Plate. Let us consider a sym- J 
metrical case of bending in which a uniform 
load q is distributed over the shaded rectan- 
gle (Fig. 69) with the sides u and v. 

We begin by developing the load in the series 



__J^L_- 



v 



'2 



Fig. 69 



2 V 



. mirx f Ha+u) . mir* ,, 
- , sin / a sin a£ 

m= 1 



-M) 



a 



= 4g y 

7T Z^ 



m = 1,3,5, . , 



(_l)(m-l)/2 m mirU # m7rX 

sin — r — sin (a) 

m 2a a 



136 THEORY OF PLATES AND SHELLS 

which represents the load for the portion prst of the plate. The corre- 
sponding deflection of this portion of the plate is governed by the differ- 
ential equation (103), which becomes 

00 

d 4 w d 4 w , d A w 4g V (-I)*™- 1 )' 2 . rmru . rmrx /1A 

t~t + 2 ^ 2 * 2 + in = Hk / ~ sin -s — sin (6) 

3x 4 dx 2 dy 2 dy 4 ttD Zv w 2a a v ' 

m = 1,3,5,... 

Let us again take the deflection in the form 

w = Wi + w 2 (c) 

where Wi is a particular solution of Eq. (6), independent of the variable y 9 
that is, satisfying the equation 

00 

d 4 Wi 4a V* (— l) (m - 1)/2 . rmru . rrnrx 



M^i = 4g V (-1) 
3a: 4 ttD Z/ ? 



sin -= — sin 



3a; 4 7rZ> Z-/ m 2a a 

m= 1,3,5. 

Integrating this latter equation with respect to x, we obtain 

oo 

4#a 4 V ( — l) Ctn ~ 1)/2 . rmru . rmrx /JX 

Wi = -4ri / " S sm -o— sm W 

7r 5 D jL/ . m b 2a a v • 

m « 1,3,5, ... 

Then w 2 must be a solution of Eq. (137) (page 114). Choosing the form 
(136) for this solution and keeping in the expression (138) for Y m only 
even functions of y, because of the symmetry of the deflection surface 
with respect to the x axis, we have, by Eq. (c), 

00 

w = / ( a m + A m cosh ^M\+. B m ^^ sinh ^^ J sin ^^ (e) 
iL/ \ a a a / a 

w= 1,3,5, ... 

in which, this time, 

am = 4£^ ( _ 1) ( m -x>, 2sin !^ (/) 

TT b m b D 2a . 

Equation (e) represents deflections of the portion prst of the plate. 

Considering now the unloaded portion of the plate below the line ts 
we can take the deflection surface in the form 



tf - V ( A ' m cosh ^ + B' m ^ sinh = 
Z-/ \ a a a 

w= 1,3,5, ... 

+ Cl sinh ™* + jy. ™v C osh ^ sin ^ (g) 
a m a a J a w 

It is now necessary to choose the constants A m , B m , . . . , D' m in the 



SIMPLY SUPPORTED RECTANGULAR PLATES 137 

series (e) and (g) in such a manner as to satisfy the boundary conditions 
a t y = 0/2 and the continuity conditions along the line ts. To repre- 
sent these conditions in a simpler form, let us introduce the notation 

rmrb rrnrv /tn 

«- = "25T 7m = ~4^~ W 

The geometric conditions along the line ts require that 

dw dw' , v ,.. 

and — = -=— f or ?/ = k (*) 



w = w 



ch/ 3y J 2 



(*) 



Furthermore, since there are no concentrated forces applied along the 
line ts, the bending moments M y and the shearing forces Q v must be 
continuous along this line. Observing Eqs. (t) these latter conditions 
can be written down in the form 

d 2 w d 2 w' , 3 3 w 3V . y ,.* 

-r-r = -7-5- an d ^f = -TT lor y = jr (J) 

Substituting expressions (e) and (g) in Eqs. (i) and (j) and using notation 
(7i), we can represent these equations in the following form: 

{A m - A J cosh 2 7m + (£ m - J B;)2 7 m sinh 2 7m 

- C^ sinh 2 7m - £>;„2 7m cosh 2 7m + a m = 
(A m - A'J sinh 27™ + (B m - K)(sinh 2 7m + 2 7m cosh 2 7m ) 

- C« cosh 2 7m - jD^cosh 2 7m + 2 7m sinh 2 7ra ) = 
{A m - A'J cosh 2 7ro + {B m - B'J{2 cosh 2 7m + 2 7m sinh 2 7m ) 

- C' m sinh 2 7m - D' m (2 sinh 2 7m + 2 7m cosh 2 7m ) = 
{A m - AJ sinh 2 7m + {B m - B' n ){3 sinh 2 7m + 2 7m cosh 2 7m ) 

- C' m cosh 2 7m - D' m {3 cosh 2 7m + 2 7m sinh 2 7m ) = 

From these equations we find 

A m — A' m = a m {y m sinh 2 7m — cosh 2 7m ) 
B m - B' m - y cosh 2 7m 

C' m = a m {y m cosh 2 7m — sinh 2 7m ) 
D' m = 2= sinh 2 7m 

To these four equations, containing six constants A m , . . . , D' m , we add 
two equations representing the boundary conditions at the edge y = 6/2. 
Substituting expression {g) in the conditions w' = 0, dW/dy* = at 
y = b/2 we obtain 

A' m cosh a m + B' m a m sinh a n + C' m sinh a m + D' m a cosh a m = ^ 

5' cosh a m -f- D' m sinh a m = 



(0 



138 THEORY OF PLATES AND SHELLS 

Equations (m), together with Eqs. (Z), yield the constants 

A m = - ttm 
m cosh a m 

cosh (a m - 27m) + y m sinh (a m - 27 m ) + a m —^ ^a I ^' 
Substituting these and expression (/) in Eq. (e), we obtain 






(-l) (m_1)/2 • rmru 
w = tS-e 7 *-= sm 



m 6 2a 

m= 1,3,5, ... 



' cosh « 

1- 



cosh 



«„ 



cosh (a m - 27m) + 7m sinh (a m - 27m) + «m Neosho" 



cosh (o: m — 27 m ) miry . , raTry 



2 cosh a m a a 



- sm (142) 



From this equation the deflection at any point of the loaded portion of 
the plate can be calculated. 

In the particular case where u = a and v = b we have, from Eqs. (h), 
7m = «m/2. Expressions (n) become 



^ = ~d^( 1 + f tanh "-) 



R _ a 

-Dm — 



2 cosh a„ 



and Eq. (142) coincides with Eq. (139) (page 116) derived for a uni- 
formly loaded rectangular plate. 

The maximum deflection of the plate is at the center and is obtained 
by substituting y = 0, x = a/2 in formula (142), which gives 



4ga 4 V* 1 . rrnru L 1 

Wmax = -TR-T / £ Sin -75 1 1 r 

Dtt 5 Lj m b 2a I cosh a m 

m= 1,3,5, ... 

Tcosh (a m - 2 T m) + 7m sinh (a m -27m) + ^ I'c^sh^ l) (143) 

As a particular example let us consider the case where u = a and v is 
very small. This case represents a uniform distribution of load along 
the x axis. Considering y m as small in Eq. (143) and retaining only small 
terms of the first order, we obtain, using the notation qv = g , 

g a 3 V (-l) (m ~ 1)/2 A u «m \ nAA , 



m« 1,3,5, 



SIMPLY SUPPORTED RECTANGULAR PLATES 

For a square plate this equation gives 



139 



W t 



0.00674 ^g! 



In the general case the maximum deflection can be represented in the form 



Wmax = <* 



= a 



goa 3 
D 

D 



for a < b 
for a > b 



Several values of the coefficient a are given in Table 19. 

Table 19. Deflections of Simply Supported Rectangular Plates 
Uniformly Loaded along the Axis of Symmetry Parallel to 

the Dimension a 
tiWx = aq a z /D 



b/a 

a. 


2 
0.001)87 


1.5 
0.00911 


1.4 

0.00882 


1.3 

0.00844 


1.2 
0.00799 


1.1 
0.00742 


1.0 

0.00674 


a/b 

a 


1.1 

0.00802 


1.2 

0.00926 


1.3 
0.01042 


1.4 
0.01151 


1.5 
0.01251 


2.0 
0.01629 


0.02083 



Returning to the general case where v is not necessarily small and u 
may have any value, the expressions for the bending moments M x and 
My can be derived by using Eq. (142). The maximum values of these 

Table 20. Coefficients /3 for (M*) ma x in Simply Supported Partially 

Loaded Square Plates 
v = 0.3 



u/a = 





0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


1.0 


v/a 


Coefficients P in the expression (M x ) max = &P 




0.1 

0.2 

0.3 

0.4 

0.5 

0.6 
0.7 
0.8 
0.9 
1.0 


OO 

0.378 
0.308 
0.262 
0.232 
0.208 

0.188 
170 
155 
141 
0.127 


0.321 
0.284 
0.254 
0.225 
0.203 
0.185 

0.168 
0.153 
0.140 
0.127 
0.115 


0.251 
0.232 
0.214 
0.195 
0.179 
0.164 

0.150 
0.137 
0.126 
0.115 
0.105 


0.209 
0.197 
0.184 
0.168 
0.158 
0.146 

0.135 
0.124 
0.114 
0.104 
0.095 


0.180 
0.170 
0.161 
0.151 
0.141 
0.131 

0.121 
0.112 
0.103 
0.094 
0.086 


0.158 
0.150 
0.142 
0.134 
0.126 
0.116 

0.109 
0.101 
0.094 
0.086 
0.078 


0.141 
0.134 
0.127 
0.120 
0.113 
0.106 

0.099 
0.091 
0.085 
0.078 
0.071 


0.125 
0.120 
0.114 
0.108 
0.102 
0.096 

0.090 
0.083 
0.077 
0.070 
0.064 


0.112 
0.108 
0.103 
0.098 
0.092 
0.087 

0.081 
0.076 
0.070 
0.064 
0.058 


0.102 
0.098 
0.093 
0.088 
0.084 
0.079 

0.074 
0.069 
0.063 
0.058 
0.053 


0.092 
0.088 
0.084 
0.080 
0.076 
0.071 

0.067 
0.062 
0.057 
0.053 
0.048 



140 



THEORY OF PLATES AND SHELLS 



Table 21. Coefficients p and Pi for (M x ) m&x and (Af„) ma x in Partialis 

Loaded Rectangular Plates with b = 1.4a 

v = 0.3 



u/a = 





0.2 


0.4 


0.6 


0.8 


1.0 





0.2 


0.4 


0.6 


0.8 


1.0 


v/a 


Coefficient /3 in the expression 

(I,)m.x = (3P 


Coefficient /3i in the expression 

(MJmax - PlP 




0.2 

0.4 

0.6 

0.8 

1.0 

1.2 

1.4 


00 

0.332 
0.261 
0.219 
0.187 
0.162 
0.141 
0.123 


0.276 
0.239 
0.207 
0.181 
0.158 
0.139 
0.122 
0.106 


0.208 
0.186 
0.168 
0.161 
0.134 
0.118 
0.104 
0.091 


0.163 
0.162 
0.138 
0.126 
0.112 
0.100 
0.089 
0.077 


0.134 
0.125 
0.115 
0.105 
0.094 
0.084 
0.075 
0.065 


0.110 
0.103 
0.095 
0.086 
0.078 
0.070 
0.062 
0.054 


00 

0.246 
0.177 
0.138 
0.112 
0.093 
0.079 
0.068 


0.299 
0.208 
0.157 
0.125 
0.102 
0.085 
0.072 
0.062 


0.230 
0.175 
0.138 
0.111 
0.091 
0.077 
0.065 
0.056 


0.183 
0.147 
0.119 
0.097 
0.080 
0.068 
0.058 
0.050 


0.151 
0.124 
0.101 
0.083 
0.069 
0.058 
0.050 
0.043 


0.125 
0.102 
0.083 
0.069 
0.058 
0.049 
0.042 
0.036 



Table 22. Coefficients p and pi for (M x ) mVkJi and (M y ) ma x in Partially 

Loaded Rectangular Plates with b = 2a 

v - 0.3 



u/a = 





0.2 


0.4 


0.6 


0.8 


1.0 





0.2 


0.4 


0.6 


0.8 


1.0 


v/a 


Coefficient /3 in expression 
(M*) ma x = (3P 


Coefficient /3i in expression 

(My)*** = PiP 




0.2 

0.4 

0.6 

0.8 

1.0 

1.2 
1.4 
1.6 
1.8 
2.0 


00 

0.347 
0.275 
0.233 
0.203 
0.179 

0.161 
0.144 
0.130 
0.118 
0.107 


0.289 
0.252 
0.221 
0.195 
0.174 
0.155 

0.141 
0.127 
0.115 
0.104 
0.094 


0.220 
0.199 
0.181 
0.164 
0.148 
0.134 

0.122 
0.111 
0.101 
0.091 
0.083 


0.175 
0.163 
0.150 
0.138 
0.126 
0.115 

0.105 
0.096 
0.087 
0.079 
0.072 


0.144 
0.135 
0.125 
0.115 
0.106 
0.097 

0.089 
0.081 
0.074 
0.067 
0.061 


0.118 
0.111 
0.103 
0.095 
0.088 
0.080 

0.074 
0.068 
0.062 
0.056 
0.051 


oo 

0.242 
0.172 
0.133 
0.107 
0.089 

0.074 
0.064 
0.056 
0.049 
0.044 


0.294 
0.203 
0.152 
0.120 
0.097 
0.081 

0.068 
0.058 
0.051 
0.045 
0.041 


0.225 
0.170 
0.133 
0.106 
0.087 
0.073 

0.061 
0.052 
0.046 
0.041 
0.037 


0.179 
0.143 
0.114 
0.093 
0.076 
0.064 

0.054 
0.046 
0.040 
0.036 
0.032 


0.148 
0.120 
0.097 
0.079 
0.065 
0.055 

0.046 
0.040 
0.035 
0.031 
0.028 


0.122 
0.099 
0.081 
0.066 
0.054 
0.046 

0.039 
0.033 
0.029 
0.026 
0.023 



moments occur at the center of the plate and can be represented by the 
formulas 

(ilfx)max = Puvq = 0P (Ar„)»u* = Piiwq = piP 

where P = uvq is the total load. The values of the numerical factors j8 
for a square plate and for various sizes of the loaded rectangle are given 
in Table 20. The coefficients /3i can also be obtained from this table by 
interchanging the positions of the letters u and v. 



SIMPLY SUPPORTED RECTANGULAR PLATES 141 

The numerical factors jS and /3 X for plates with the ratios & = \Aa and 
b = 2a are given in Tables 21 and 22, respectively. 1 

34. Concentrated Load on a Simply Supported Rectangular Plate. 
Using Navier's method an expression in double-series form has been 
obtained in Art. 29 for the deflection of a plate carrying a single load P 
at some given point x = £, y = *? (Fig. 70). To obtain an equivalent 
solution in the form of a simple series we begin by representing the Navier 
solution (133) in the following manner: 



w = 



7r 4 aD 



00 

I 

m— 1 



„ . mw£ . rmrx 
bm sin sin 



a 



a 



(a) 



the coefficient £ m being given by 



S n 



717T77 . n7T/7 

1 sin -»,-- 
o 



... (^+-y 



(6) 



1 

* A 


i 
i 

i 
i 
i 
i 

b 
i 
i 
i 

i 
i 
i 

♦ 


< a -> 





Introducing the notation 

nir(y - ij) 
V~"^ cos 

s„ 



Fig. 70 



i 



Zj(m?b*. A 2 ~ 2L/(m*W, *\ 



oo 

2 



cos 



wx(if + y) 



(c) 



we can also represent expression (6) in the form 
To evaluate the sums (c) we use the known series 



00 

I 

n = l 



cos nz __ __ 1 ir cosh <*(7r — 2) 

a 2 + n 2 2a 2 2a sinh 7ra 



(<*) 



(e) 



which holds for < z < 2tt and which we regard, first of all, as a func- 
tion S(a) of a. Differentiation of the left-hand side of Eq. (e) with 
respect to a gives 



dS( 



- - - 2 ° 2 r 

71 = ] 



cos nz 



+ n 2 ) 



2\2 



(/) 



After differentiating also the right-hand side of Eq. (e) and substituting 

1 The values of M x and M y for various ratios a/b, u/a, and v/b are also given in the 
form of curves by G. Pigeaud, Ann. ponts et chausstes, 1929. See also Art. 37 of this 
book. 



142 THEORY OF PLATES AND SHELLS 

the result in Eq. (/), we conclude that 



oo 

X 

n = l 



cos nz 
(a 2 + W 2 ) 2 



1 dS(a) 
2a da 



1 _7r_ cosh a(ir — z) 
2a 4 4a 3 sinh ira 



w(t — g) slnh a(7r — g) , tt 2 cosh a(ir — g) cosh wa 
4a 2 sinh 7ra 4a 2 



sinh 2 ira 



(9) 



Now, to obtain the values of the sums (c) we have to put, in Eq. (g) t 
firsts = (ic/b)(y — r)), then z = {j/b){y + rj) and, in addition, a = m6/a. 
Using these values for substitution in Eqs. (d) and (a) we arrive, finally, 
at the following expression for the deflection of the plate: 



V) 



- % i ( i + *• -* *- - -t c ° th *? - x ~* t) 



sinh -r 1 smh — r^- sin sin 



a 



a 



m 3 sinh /3 n 



(145) 



in which 



]8m = 



rrnrb 



a 



2/i = b- ?/ 



and 



V >rj 




In the case oi y < rj the quantity t/i must be 
replaced by ?/ and the quantity q by rji = b — t\ y in 
using expression (145). 

Let us consider more closely the particular case 
of a load P concentrated at a point A on the axis 
of symmetry of the plate, which may be used as the 
x axis (Fig. 71). With rj = 6/2 and the notation 



<x n 



Fig. 71 



rrnrb 
~2a 



2 



W 



the general expression (145) for the deflection of the plate becomes 

00 

w = ^-ifi) X U 1 + a » tanh "•») sinh y ( & ~~ 2y ) 



m= 1 



-2=(&-2j,)cosh2=(&-2y)] 



. ra7r£ . rmr:: 
sin sm 



a 



a 



m 3 cosh a« 



(146) 



which is valid for y > 0, that is, below the x axis in Fig. 71. Putting, 
in particular, y = we obtain the deflection of the plate along the x axis 
in the form 



(w)y-0 



SIMPLY SUPPORTED RECTANGULAR PLATES 143 

(i) 



Pa 
2v 3 D 



d 2^ ( tanh °~ ~ ^S^) 

m— 1 



sin 



a 



sin 



a 



m° 



This series converges rapidly, and the first few terms give the deflections 
with sufficient accuracy. In the case of a load P applied at the center of 
the plate, the maximum deflection, which is at the center, is obtained by 
substituting x = £ = a/2 in expression (i). In this way we arrive at the 
result 



w n 



Pa2 V ! (* u *- N \ Pa2 nA7^ 

27r 3 D jLj m*\ cosh 2 a m / D 

m=l 

Values of the numerical factor a for various values of the ratio b/a are 
given in Table 23. 

Table 23. Factor a for Deflection (147) of a Centrally Loaded 

Rectangular Plate 



b/a = 



1.0 



1.1 



1.2 



1.4 



1.6 



1.8 



2.0 



3.0 



a = 0.01160 0.01265 0.01353 0.01484 0.01570 0.01620 0. 01651 0.01G90 0.01G95 

It is seen that the maximum deflection rapidly approaches that of an 
infinitely long plate 1 as the length of the plate increases. The compari- 
son of the maximum deflection of a square plate with that of a centrally 
loaded circular plate inscribed in the square (see page 68) indicates that 
the deflection of the circular plate is larger than that of the corresponding 
square plate. This result may be attributed to the action of the reactive 
forces concentrated at the corners of the square plate which have the 
tendency to produce deflection of the plate convex upward. 

The calculation of bending moments is discussed in Arts. 35 and 37. 

35. Bending Moments in a Simply Supported Rectangular Plate with 
a Concentrated Load. To determine the bending moments along the 
central axis y == of the plate loaded according to Fig. 71 we calculate 
the second derivatives of expression (146), which become 



S. rrnr% 
a /, , a m \ . rriTX 
[ tanh a m r^ J sin 
m \ cosh 2 a m ) a 



m~\ 



/dhv\ = __ 






W7r£ 



sin 



rmrx 



a 



771=1 



1 The deflection of plates by a concentrated load was investigated experimentally 
by M. Bergstrasser; see Forschungsarb., vol. 302, Berlin, 1928; see also the report of 
N. M. Newmark and H. A. Lepper, Univ. Illinois Bull., vol. 36, no. 84, 1939. 



144 THEORY OF PLATES AND SHELLS 

Substituting these derivatives into expressions (101) for the bending 
moments, we obtain 



^-^ [(1 + ,) tanh a m - fi^M sin ™ 
m I cosh 2 a m J a 






When & is very large in Comparison with a, we can put 



rrnrx 

(a) 
mrx 



tanh <* m « 1 — ^— » 
cosh 2 a m 

00 

Tlwwi /7Lf >i _ /UN (1 + V)P V 1 . mx£ . 1UWX . 

men (mJh = MJ^o = - — — / — sin — - sm (6) 

2?r Z-/ m a a v y 

m^ 1 

This series does not converge rapidly enough for a satisfactory calcu- 
lation of the moments in the vicinity of the point of application of the 
load P, so it is necessary to derive another form of representation of the 
moments near that point. From the discussion of bending of a circular 
plate by a force applied at the center (see Art. 19) we know that the 

shearing forces and bending moments become infi- 
nitely large at the point of application of the load. 
We have similar conditions also in the case of a rec- 
tangular plate. The stress distribution within a 
circle of small radius with its center at the point of 
application of the load is substantially the same as 
that near the center of a centrally loaded circular 
plate. The bending stress at a point within this cir- 
cle may be considered as consisting of two parts : one 
is the same as that in the case of a centrally loaded 
circular plate of radius a, and the other represents the 
difference between the stresses in a circular and those 
jr IG# 72 in a rectangular plate. As the distance r between 

the point of application of the load and the point 
under consideration becomes smaller and smaller, the first part of the 
stresses varies as log (a/r) and becomes infinite at the center, whereas 
the second part, representing the effect of the difference in the boundary 
conditions of the two plates, remains continuous. 

To obtain the expressions for bending moments in the vicinity of the 
point of application of the load we begin with the simpler case of an 
infinitely long plate (Fig. 72). The deflection of such a plate can readily 




SIMPLY SUPPORTED RECTANGULAR PLATES 145 

be derived from expression (146) by increasing the length of the side 6, 
and consequently the quantity a m = mirb/2a, indefinitely, i.e., by putting 

tanh a m « 1 cosh a m « \e am 

sinh y (6 - 2y) » cosh y (6 - 2y) « ~ c («-/«(«if) 

Substituting this into Eq. (146) the required deflection of the simply 
supported strip carrying a concentrated load P at x = £, y = becomes 1 

00 

Pa 2 V* 1 . ra7r£ . rrnrx ( , , m7r2/\ , ,.. „™ 

= TTTn / ~ i sm " sm I 1 H " ) e~ m,r ^ /a (148) 



w 

'Ztt*U 



which holds for y > 0, that is, below the x axis (Fig. 72). 

The corresponding expressions for the bending moments and the twist- 
ing moment are readily obtained by means of Eqs. (101) and (102). We 
have 



M x = pr > - sm — - sm 1 + *> + (1 - v) — - e^ mrvla 

27r Z./ ma a L a J 

m~ 1 

00 

M„ = 7T- > - Sin Sin 1 + V - (1 - J>) 2 6 -m*rv/« Q49) 

2tt L*/ m a a[ a J 

00 
ikf^ = - ~ y(\ - *>) > sm cos e - m * vla 

w= 1 

Once again using the quantity M = (M, + M y )/(1 + j>) introduced 
on page 92, we have 

00 

jlf = _-2)( + ) = - > — sin — - sm e -™*v/« (J5()) 

\dx 2 dy 2 J ir Zv m a a 



m=l 



The moments (149) can be expressed now in terms of the function M in 
the following simple manner : 

M v = ±[ (1 + v)M + (1 - v)y ^ j (151) 

1 This important case of bending of a plate has been discussed in detail by A. N&dai: 
see his book "Elasiisehe Platten," pp. 78-109, Berlin, 1925. 



146 THEORY OF PLATES AND SHELLS 

Summing up the series (150), we obtain the expression 1 

, iry tt(x + £) 

cosh — — cos — — — - 

M = f- log j^—pr (152) 

cosh — — cos — — 

a a 

and, using Eqs. (151), we are able now to represent the moments of the 
infinitely long plate in a closed form. Observing, furthermore, that 
AAw = everywhere, except at the point {x = £, y = 0) of the appli- 
cation of the load, we conclude that the function M = — D Aw satisfies 
(except at the above-mentioned point) the equation AM = 0. By virtue 
of the second of the equations (111) the boundary condition M = along 
the edges x = and x = a is also satisfied by the function M. 

For the points along the x axis Eqs. (151) yield M x = M y and therefore 

(Af,)^o = {M y ) y =« = (M) ye=0 ^ (c) 

Using Eqs. (c) and P]q. (152) in the particular case of a load applied at 
the center axis of the strip, £ = a/2, we obtain 

. TTX 

pn , x 1 + sin — 

(M x ) v ^ = {M y ) y ^ = rU 8 ; " J log £ (d) 

1 — sin — 
a 

a result which also can be obtained by summation of the series (6). 

Now let us return to the calculation of bending moments for points 
which are close to the point of application of the load but not necessarily 
on the x axis. In this case the quantities (x — £) and y are small and, 
using expression (152), we can put 



cos _v___ « i ^__ cosh- « 1+2^ 



jw « i - y -v«- *'" cosh^ « 1 + f #2 
a 2a 2 a 

Thus we arrive at the result 

1 — cos — 

M = ^ log \ , gg i , " 2 (*-*) 2 

"•"2a 2 ■ 2a 2 

_ /2a sin— \ „ 2o sin — 

= £ log I ± ) = P i og £L (153) 

4?r & \ ?rr / 2tt & 7rr 

1 See, for instance, W. Magnus and F. Oberhettinger, "Formeln und Satze fur die 
speziellen Funktionen der mathematischen Physik," 2d ed., p. 214, Berlin, 1948. 



SIMPLY SUPPORTED RECTANGULAR PLATES 



147 



in which 



r = V(x — £) 2 + y 2 

represents the distance of the point under consideration from the point 
q{ application of the load P. Now, using expression (153) for substi- 
tution in Eqs. (151) we obtain the following expressions, valid for points 
in the vicinity of the concentrated load : 



(154) 



It is interesting to compare this result with that for a centrally loaded, 
simply supported circular plate (see Art. 19). Taking a radius r under 
an angle a to the x axis, we find, from Eqs. (90) and (91), for a circular 
plate 



M x = \ 


_. 2a sin — ,., \r> -> 
r , , n P i a , (1 - v)Py* 


M y = I 


o • ** 



P a 

M x = M n cos 2 a + M, sin 2 « = — (1 + v) log - + (1 

M y = 7l/ n sin 2 a + M 4 cos 2 a = £- (1 + v) log * + (1 



x p ?/ 2 



(e) 



v) 



4tt r 2 



The first terms of expressions (154) and (<?) will coincide if we take the 
outer radius of the circular plate equal to 

2a . tt£ 

— sin — 

7r a 

Under this condition the moments M x are the same for both cases. The 
moment M y for the long rectangular plate is obtained from that of the 
circular plate by subtraction of the constant quantity 1 (1 — v)P/4t. 
From this it can be concluded that in a long rectangular plate the stress 
distribution around the point of application of the load is obtained by 
superposing on the stresses of a centrally loaded circular plate with 
radius (2a/ir) sin (irf/a) a simple bending produced by the moments 
My = -(1 - v)P/4*. 

It may be assumed that the same relation between the moments of 
circular and long rectangular plates also holds in the case of a load P 
uniformly distributed over a circular area of small radius c. In such a 
case, for the center of a circular plate we obtain from Eq. (83), by neg- 
lecting the term containing c 2 , 

M max = £ [(1 + ,)log^+l] 

47T |_ C j 

1 We observe that x 2 = r 2 — y 2 . 



148 



THEORY OP PLATES AND SHELLS 



Hence at the center of the loaded circular area of a long rectangular 
plate we obtain from Eqs. (154) 



M x = 4- 



M v =4- 



p 


2a sin — 


At 
P 


2a sin — 
n i w^ ino- a 1 1 


4x 


^1 1 V) lOg "11 
7TC 



(155) 



(1 - v)P 
4tt 



From this comparison of a long rectangular plate with a circular plate 
it may be concluded that all information regarding the local stresses at 
the point of application of the load P, derived for a circular plate by- 
using the thick-plate theory (see Art. 19), can also be applied in the case 
of a long rectangular plate. 

When the plate is not very long, Eqs. (a) should be used instead of 
Eq. (6) in the calculation of the moments M x and M y along the x axis. 
Since tanh <x m approaches unity rapidly and cosh a m becomes a large 
number when m increases, the differences between the sums of series (a) 
and the sum of series (6) can easily be calculated, and the moments M x 
and My along the x axis and close to the point of application of the load 
can be represented in the following form: 



M* = 



d + 



+ y)p y i 

2t L*l TTi 



. rtor£ . rrnrx . P 

sm sin h 7i t~ 

m a a 4w 



m=l 



tt£ 



oc 

(i + o^ V i • w * 



47T 10g Trr ^ 7i 4tt (i 56 ) 
miri- . rrnrx , P 

P(l + v) , ^ Sln f , P 

in which 71 and 72 are numerical factors the magnitudes of which depend 
on the ratio b/a and the position of the load on the x axis. Several values 
of these factors for the case of central application of the load are given in 
Table 24. 

Again the stress distribution near the point of application of the 
load is substantially the same as for a centrally loaded circular plate 
of radius (2a/ir) sin (v^/a). To get the bending moments M x and M y 
near the load we have only to superpose on the moments of the 



SIMPLY SUPPORTED RECTANGULAR PLATES 
Table 24. Factors 71 and 72 in Eqs. (156) 



149 



b/a 


1.0 


1.2 


1.4 


1.6 


1.8 


2.0 


OC 


T2 


-0.565 
+0.135 


-0.350 
+0.115 


-0.211 
+0.085 


-0.125 
+0.057 


-0.073 
+0.037 


-0.042 
+0.023 







circular plate the uniform bending by the moments M' x = yiP/iir and 
jlf ' = — (1 — j> — 7 2 )P/47r. Assuming that this conclusion holds also 
when the load P is uniformly distributed over a circle of a small radius c, 
we obtain for the center of the circle 



M x = 4- 



P 

4tt 



My = 



P_ 

4ir 



2a sin — 

n I iA lno- a 4- 1 


, TiP 
- (1 - v- 




7TC 

2a sin ^ 

(-[ I ,A l no . a _|_ 1 


"T 2 )£ 


U 1 ^ 10 S ^g ^ 1 



(157) 



Just as in the case of a distributed load, reactive forces acting down- 
ward and considerable clamping moments are produced by concentrated 
loads at the corners of a rectangular plate. The corner reactions 

R = nP (/) 

due to a central load P are given in Table 25 by the numerical values of 
the factor n, whereas the clamping moments have the value of — R/2 
(see page 85). The computation of the values of R has been carried out 
by a simple method which will be described in Art. 36. 

Table 25. Numerical Factor n for Reactive Forces R at the Corners 
of Simply Supported Rectangular Plates under Central Load 

v = 0.3 



b/a = 


1.0 


1.2 


1.4 


1.6 


1.8 


2.0 


3.0 


00 


n = 


0.1219 


0.1162 


0.1034 


0.0884 


0.0735 


0.0600 


0.0180 






The distribution of the bending moments and reactive pressures in the 
particular case of a square plate with a central load is shown in Fig. 73. 
The dashed portion of the curves holds for a uniform distribution of the 
load P over the shadowed circular area with a radius of c = 0.05a. 

36. Rectangular Plates of Infinite Length with Simply Supported 
Edges. In our foregoing discussions infinitely long plates have been 
considered in several cases. The deflections and moments in such plates 
were usually obtained from the corresponding solutions for a finite plate 
by letting the length of the plate increase indefinitely. In some cases 



150 



THEORY OF PLATES AND SHELLS 



v.,-3? 




it is advantageous to obtain solutions for an infinitely long plate first 
and combine them in such a way as to obtain the solution for a finite 
plate. Several examples of this method of solution will be given in this 
article. We begin with the case of an infinitely long plate of width a 
loaded along the x axis as shown in Fig. 74. Since the deflection surface 
is symmetrical with respect to the x axis, we need consider only the por- 
tion of the plate corresponding to positive values of y in our further dis- 
cussion. Since the load is distributed only along 
the x axis, the deflection w of the plate satisfies the 
equation 

d 4 w 




d 4 w d 4 w 

dx 4 + d^Jy" 1 



+ 



dy 4 



= 



(a) 



We take the solution of this equation in the form 



w 



Fig. 74 



00 

■2 

7/1=1 



Y m sin 



rrnrx 
a 



(b) 



which satisfies the boundary conditions along the simply supported 
longitudinal edges of the plate. To satisfy Eq. (a), functions Y m must 
be chosen so as to satisfy the equation 



yiv 



* Y" + !?r. = o 



a* 



a* 



SIMPLY SUPPORTED RECTANGULAR PLATES 151 

Taking the solution of this equation in the form 

Y m = A m e™« ,a + B m ™— y - e™y>" + C m e-» **> /a + D m ^ er™ yla (c) 

a a 

and observing that the deflections and their derivatives approach zero 
at a great distance from the x axis, it may be concluded that the con- 
stants A m and B m should be taken equal to zero. Hence solution (b) 
can be represented as follows: 



w = £ (c m + D m ^) e—/. sin ^ (d) 



m = 1 

From the condition of symmetry we have 



( 



This condition is satisfied by taking C m = A» in expression (d). Then 



771= 1 



w = > C m ( 1 + = ) 6— »/. S in ^ (e) 



The constants C w can be readily calculated in each particular case pro- 
vided the load distribution along the x axis is given. 

As an example, assume that the load is uniformly distributed along 
the entire width of the plate. The intensity of loading can then be 
represented by the following trigonometric series: 

00 

4 V* 1 . rn-wx 

q =* - Qo / — sm 

vi— 1,3,5, . . . 

in which q is the load per unit length. Since the load is equally divided 
between the two halves of the plate, we see that 

00 

m=l,3,5... 

Substituting expression (c) for w, we obtain 

00 00 

• — 5— / 6mW 3 sm = -— / — sm 

771 = 1 771= 1,3,0, . . . 

from which 

Cm = tt^ — t where ra = 1 , 3, 5. . ., r 
Drm* 



152 THEORY OF PLATES AND SHELLS 

00 

Hence w = 5** Y ± ( 1 + ^) e— '. sin ™ (f/) 

7r 4 Z> jLi m A \ a / a * 

m- 1,3,5,... 

The deflection is a maximum at the center of the plate (x = a/2, y = 0), 
where 

Wraax " 7T 4 D Z/ m 4 1,536Z) w 

m= 1,3,5, ... 

The same result can be obtained by setting tanh a m = 1 and cosh a m = oo 
in Eq. (144) (see page 138). 

As another example of the application of solution (e), consider a load 
of length u uniformly distributed along a portion of the x axis (Fig. 74). 
Representing this load distribution by a trigonometric series, we obtain 



4go V 1 • W7r£ . rrnru . mwx 

a = -2- / — sin sin -pr— sm 

7r Z-/ ra a 2a a 

TO 3 * 1 



where qn is the intensity of the load along the loaded portion of the x axis. 
The equation for determining the constants C m , corresponding to Eq. (/), 
is 

00 

~ d (d 2 w . d 2 w\ 2qo V s 1 . W7r£ . rrnru . rrnrx 

D — I — r + — 5 1 = — — / — Sin 1 Sin "tt— Sin 



dy\dx 2 dy 2 / ya o w L*j m a 2a a 

m=l 



dy 2 /y-o t Li m 

m=l 

Substituting expression (e) for w, we obtain 

00 00 

2Dt z Y* „ 9 . ?n7ra: 2g \^ 1 . rrnc% . ra^m . rrnrx 

— r- / Cmm 3 sm = - 3 - / — sm — ™ 

a 3 Z-/ a ir Lj m a 

m 

from which 



sm -a- — sm 
2a a 

m = 1 m™ 1 



~ aoa 3 . rmr^ . m^ra 

C m = * 4 sm sm -x— 

7T 4 Dm 4 a 2a 

Expression (e) for the deflections then becomes 

oo 

<7oa 3 V* 1 • W7r £ • rrnru ( ^ , rrnry\ „. . rrnrx ,. N 

w = ^t^: / — A sm — - sm -^r— ( 1 H ) e- mrv/a sm (i) 

tc 4 D jLi m 4 a 2a \ a / a w 

m = l 

The particular case of a concentrated force applied at a distance £ from 
the origin is obtained by making the length u of the loaded portion of the 
x axis infinitely small. Substituting 



SIMPLY SUPPORTED RECTANGULAR PLATES 



153 



qou 
in Eq. (i), we obtain 



and 



sin 



TUTU 



a 



rmru 
a 



w = 




. rrnr% / \ , rmry\ 

sin ( 1 H J e~ 

a \ a / 



mvyla c\r\ 



rrnrx 
a 



(158) 



*H 



?£3 



^ 



teseS 



-I 



V 

,2 



dV 



* 2 



an expression that coincides with expression (148) of the preceding article. 

We can obtain various other cases of loading by 
integrating expression (i) for the deflection of a long 
plate under a load distributed along a portion u of the 
x axis. As an example, consider the case of a load 
of intensity q uniformly distributed over a rectangle 
with sides equal to u and v (shown shaded in Fig. 75). 
Taking an infinitesimal element of a load of magni- 
tude qu dt] at a distance rj from the x axis, the corre- 
sponding deflection produced by this load at points 
with y > 7] is obtained by substituting q drj for q and 
y — 7j for y in expression (i). The deflection pro- 
duced by the entire load, at points for which y 2> v/2, is now obtained by 
integration as follows: 



<r ► 



Fig. 75 



w 



__ qa* V 1 mTr 
~~ 7T 4 Z) Z/ m 4 a 

m=l 



m7r£ . m7ru . rrnrx 
sin -7T — sm 



2a 



a 



/V2 
7 -v/2 



??Z7r£ . rrnrU 
sin sin -^ — 



rrnrx 



1 + 



7717T(?/ - 77) ] 



nnr(y — v) 



a 



dn 



9a 3 V 1 - 

= 7T^ / — a S11 
7T 4 i) Zv ™ 4 

m= 1 



U) 



By a proper change of the limits of integration the deflection at points 
with y < v/2 can also be obtained. Let us consider the deflection along 
the x axis (Fig. 75). The deflection produced by the upper half of the 
load is obtained from expression (j) by substituting the quantity v/4 for 
// and for v/2. By doubling the result obtained in this way we also take 
into account the action of the lower half of the load and finally obtain 



Mh 



4qa 4 V 1 • 

~^ / r Si 

7r 5 D Zy m° 

m= 1 



m-K^t . m7ru . rrnrx 
sm sin — — sin 



a 



2a 



a 



1 



O+^VH 



(4) 



154 THEORY OF PLATES AND SHELLS 

When v — » , the load, indicated in Fig. 75, is expanded along the entire 
length of the plate, and the deflection surface is cylindrical. The corre- 
sponding deflection, from expression (/c), is 



(w) 



y*=0 



00 

= -^=r > — sin — ~ sin -7T— sin (/J 

tt 5 D Zv m 5 a 2a a 

?»= i 



Making £ = w/2 = a/2 in this expression, we obtain 



00 

4qa 4 V J_ 
7r 5 D Li m 1 



(w)v=o = 4r 7 , Z75 sin 



rrnrx 



m b a 

m = 1,3,5, . . . 



which represents the deflection curve of a uniformly loaded strip. 

The following expressions for bending moments produced by the load 
uniformly distributed along a portion u of the x axis are readily obtained 
from expression (i) for deflection w: 



(m) 



M x = -^V / — i sm "7T sm ~97T sin ~V~ 
7r 2 Z-/ m 2 a 2a a 

m = l 

00 

M« = ^V > — « sin — - sin -=— sin —— 
v 2 Zj m o 2o a 

m = l 

r 1 + v _(i_,)^ e -w«j 

These moments have their maximum values on the x axis, where 

(M*)„_ = (M y ) V=0 

oo 

g a(l + v) V^ 1 . m7r£ . rmru . m7ra: , * 

= "2 > — = sin sin -^— sm — — (n) 

* ^ m 2 a 2a a 



m=l 



In the particular case when £ = u/2 = a/2, that is, when the load is 
distributed along the entire width of the plate, 



00 

QM (1 + v) V* 1 . Wttr- 



W7ra; 
— i sm 
w 2 

ra = 1,3,5, . . , 

The maximum moment is at the center of the plate where 



SIMPLY SUPPORTED RECTANGULAR PLATES 



155 



(M 



t U = W. = ^ V ( -=^ = 0.0928 3o a(l + ,) 



ra = 1,3,5, 

When u is very small, i.e., in the case of a concentrated load, we put 



sin 



and 



2a 2a 

Then, from expression (n), we obtain 

P(l + 



go^ = P 



(M,)^ = (M y ) y= o 



2tt 



XI . m7r 
— sin — 
m a 

m= 1 



m7rf . wittj: 
sin 



a 



(o) 



which coincides with expression (6) of the preceding article and can be 
expressed also in a closed form (see page 146). 

In the case of a load q uniformly distributed over the area of a rec- 
tangle (Fig. 75), the bending moments for the portion of the plate for 
which y ;> v/2 are obtained by integration of expressions (m) as follows: 



00 

7r 2 Z-/ wi 2 a 

m= 1 



. m7ru . rrnrx 
sin — — sin 



2a 



a 



J -v/2 



1 + „ + (1 - „) 

= ^- > — sm sin -7T— sm — 

7T 2 Zy m 2 a 2a a 

m= 1 

([s+a-K'-s)]'" 

m7r 



m7r(v/ — 17) 



a 



mir{y — rj) 



drj 



rmr(2y — v) 
2a 



-) (» + 1)] 



m7r(2;/-fv) 
2a 



(159) 



_ ga V J 

m— 1 



A/„ = 



sm sin — — sm 



a 



2a 



a 



p - a - 



m7r 



(' - 1)\ 



rrnr(2y — v) 
2k 



~2va /t , / . A 

- - (1 - v) [y + ^ ) 



77Hr(2j/-fv) 1 
6 2a 1 






The moments for the portion of the plate for which y < v/2 can be calcu- 
lated in a similar manner. To obtain the moments along the x axis, we 
have only to substitute v/2 for v and v/± for y in formulas (159) and 



156 THEORY OF PLATES AND SHELLS 

double the results thus obtained. Hence 



(M x ) ym .o 



so 

4ga 2 V J^ 
tt z Zj m z 



. rrnc£ . micu . mwx 

sin — - sin — — sin 

a 2a a 



-I 



1 + (1 - v) 



mirv\ 
~4a\ 



tmrv 
~2a~ 



{My) 



•0 

_ 4qa? V J_ 

m = l 



(160) 



. m7r£ . mTU . mirx 
sin sin -^ — sin 



a 



2a 



a 



(,-[,- a -.)5].-*) 



If values of the moments at the center of the loaded rectangular area are 
required, the calculation may also be carried out by means of expressions 
(167), which will be given in Art. 37. When v is very small, Eqs. (160) 
coincide with Eq. (n) if we observe that qv must be replaced in such a 
case by 30. When v is very large, we have the deflection of the plate to a 
cylindrical surface, and Eqs. (160) become 



lMy)y^ 



4qa 2 \} 1 * m* 
= -^v" / — * sm — 

m = l 

00 

7T 3 jLi m l 



m-K% . rrnru . rrnrx 
sm — — sin 



2a 



a 



. mwZ . m^ra . mxa; 

, sm sm -jr— sm 

w jlj m* a 2a a 

m = l 



The expressions for the deflections 
and bending moments in a plate of finite 
length can be obtained from the corre- 
sponding quantities in an infinitely long 
plate by using the method of images. 1 
Let us begin with the case of a concen- 
trated force P applied on the axis of 
symmetry x of the rectangular plate 
with sides a and b in Fig. 76a. If we 
now imagine the plate prolonged in both 
the positive and the negative y direc- 
tions and loaded with a series of forces 
P applied along the line mn at a dis- 
tance b from one another and in alter- 
nate directions, as shown in Fig. 766, the deflections of such an infinitely 

1 This method was used by A. N&dai (see Z. angew. Math. Mech., vol. 2, p. 1, 1922) 
and by M. T. Huber (see Z. angew. Math. Mech., vol. 6, p. 228, 1926). 




SIMPLY SUPPORTED RECTANGULAR PLATES 157 

long plate are evidently equal to zero along the lines AiB h AB, CD, 
CiD ly .... The bending moments along the same lines are also zero, 
and we may consider the given plate ABCD as a portion of the infinitely 
long plate loaded as shown in Fig. 766. Hence the deflection and the 
stresses produced in the given plate at the point of application of the 
concentrated force can be calculated by using formulas derived for infi- 
nitely long plates. From Eq. (158) we find that the deflection produced at 
the x axis of the infinitely long plate by the load P applied at the point is 

00 

Pa 2 V 1 • ?mr £ • mirX 



V i . ™* 

/ — = sin — 
L^f ra 3 a 



Wi = o ^ 7 — 5 sin sin 

2ir z D JLj ra 3 a a 

m — l 



The two adjacent forces P applied at the distances b from the point 
(Fig. 766) produce at the x axis the deflection 



00 

Pa 2 V 1 • mT % /i i o \ -2« • m7r: 

= ^ > — ; sm (1 + 2a m )e- 2am sin -— 

7r 3 D Z-/ m z a a 

m—l 



in which, as before, 

rmrb 
~2a 



(*m = 



The forces P at the distance 26 from the point produce at the x axis 
the deflection 



00 

Pa 2 V 1 • m ^ Mia \ 
7r 3 D Z./ m s a 



rrnrx 



a 

m = 1 

and so on. The total deflection at the x axis will be given by the 
summation 

w = w± + W2 + w* + • • • (p) 

Observing that 

1 p— 2a m 

tanh a m = , , . = 1 - 2e- 2< "» + 2e- 4a - • • • 
1 -J- e~ 2am 



1 4 4e 



— 2a, n 



cosh 2 a m (e a ™ + e"""-) 2 (1 + e -2 "-) 2 

= 4 e -2«-n(i — 2e- 2or, » + 3e- 4a -» - 4e- 6a ™ + • • •) 

we can bring expression (p) into coincidence with expression (146) of 
Art. 34. 

Let us apply the method of images to the calculation of the reactive force 

R = -2M IV 



158 THEORY OF PLATES AND SHELLS 

acting at the corner D of the rectangular plate ABCD (Fig. 76) and produced by a load 
P at the center of this plate. Using Eqs. (151) and (152), we find that the general 
expression for the twisting moment of an infinitely long plate in the case of a single 
load becomes 



1 8M 

M,„ = --(!-,)!/ — 



P(l - v)y 
8a 



sin sin 

a a 



vy . ir(x - {) . wy x(a!+J) 

cosh — - cos cosh cos 

a a a a 



(9) 



Hence a load P concentrated at x = £ = a/2, y = produces at x = the twisting 
moment 

«„ - - ?<' " r) " w 

4a _ *■*/ 

cosh — 

a 

Now, putting 7/ = 6/2, 36/2, 56/2, . . . consecutively, we obtain the twisting 
moments produced by the loads ±P acting above the line DC. Taking the sum of 
these moments we obtain 



\ cosh — 
\ la 



PHI -v) l j. ±—+-^ \ (.) 

cosh — — cosh — — 
2a 2a 



To take into account the loads acting below the line DC we have to double the 
effect (s) of loads acting above the line DC in order to obtain the effect of all given 
loads. Thus we arrive at the final result 



m = 1,3,5, . • • 

As for the reactive force acting downward at the point D, and consequently at the 
other corners of the plate, it is equal to R = -2M XV , M xy being given by Eq. (t). 

The method of images can be used also when the point of application of P is not on 
the axis of symmetry (Fig. 77a). The deflections and moments can be calculated by 
introducing a system of auxiliary forees as shown in the figure and using the formulas 
derived for an infinitely long plate. If the load is distributed over a rectangle, for- 
mulas (167), which will be given in Art. 37, can be used for calculating the bending 
moments produced by actual and auxiliary loads. 

37. Bending Moments in Simply Supported Rectangular Plates under a Load 
Uniformly Distributed over the Area of a Rectangle. Let us consider once more the 
practically important case of the loading represented in Fig. 78. If we proceed as 
described in Art. 33, we find that for small values of u/a and v/b the series representing 
the bending moments at the center of the loaded area converge slowly and become 
unsuitable for numerical computation. 

In order to derive more convenient formulas 1 in this case let us introduce, in exten- 
sion of Eq. (119), the following notation: 

1 See S. Woinowsky-Krieger, Ingr.-Arch., vol. 21, p. 331, 1953. 



SIMPLY SUPPORTED RECTANGULAR PLATES 

M x + My 



M = 



N = 



M x 



1 + v 



/dhv dhv 
= - D ( 

\dx 2 dy* 



) 

(dhv Ohv\ 
~dx* ~ dy*J 



Hence 



1 - V 
M x = ^(1 + v)M + i(l - v)N 

My = |(1 + V)JI/ - i(l ~ V)N 



159 



(161) 



(162) 



At first let us consider a clamped circular plate of a radius a with a central load, 

distributed as shown in Fig. 78. The bending 
moments at the center of such a plate can be 
obtained by use of the Michell solution, for an 
eccentric single load. If u and v are small in 






■4 



-X. 2 





7^ 
i 

i 
i 

JQ|C\J 

I 



-0|C\J 



t —i 



>|C\I 

-h- 

>|C\J 

JL— 



zz 



< .H >< H > 



Fig. 77 



— a 



Fig. 78 



comparison with a , the result, evaluated by due integration of expression (197) (p. 
293), can be put in the form 



M 






2a \ 
2 + 2 log — - * J 



in which 



P 

N =- + 

4tt 



1 1 
<p — k arctan — h 7 arctan k 
k k 

\[/ = k arctan arctan k 

k k 



(a) 



(163) 



k = 



and 



u 



d = y/ u * + v 2 



For a simply supported circular plate with the same radius a as before, we have to 
add a term P/4tt to M x and Af„ (see p. 68), i.e., a term P/2tt(1 + v) to Af and nothing 
to N, so that these latter quantities become 



M 

N 



--(■■ 

4tt\ 



* + "*T-) + £oT7, 



(6) 



4x 



160 



THEORY OF PLATES AND SHELLS 



Finally, to obtain the corresponding expressions for an infinite strip (Fig. 75), 
we must assume a<> «= 2a/w sin Or£/a) and introduce an additional moment 
M v ■■ — (1 — p)P/4w (see p. 147). This latter operation changes the quantity 
M by -(1 - v)P/4tt(1 + v) and the quantity N by +P/4tt. Introducing this in 
Eqs. (6) we arrive at the result 



M 



I 4a sm — I 

-s\ 2,og -ir £+8 -7 



(164) 



AT - — (1+ *) 

47T 



The values of the factors <p and ^, depending only on the ratio v/u, are given in 
Table 26. 

Considering now the case of a rectangular plate (Fig. 78), we have only to take into 
account the effect of the auxiliary loads 1 ±P (Fig. 77) and to add this effect to the 
values (164) of M and N. The final result, in the case shown in Fig. 78, can then be 
put in the form 



^ . 4a sin 

P I a 

M = — \ 2 log — - — + X - <p k 

n - £ a. + *) 

4ir 



(165) 



where ^>, ^, d are given by expressions (163) and Table 26, and 



cosh ol v 



sin a 



w = l 






(166) 



cosh 2 a m 



sin* 



mirf 



with a m = mirb/2a. The terms X and /*, expressed by rapidly convergent series, are 
wholly independent of the dimensions u and v (and even the shape) of the loaded area. 
Their numerical values are given in Table 27. 

From Eqs. (162) we obtain the expressions for the bending moments 



OB" 



M„ = — 



4a sin — 
2 log — - + X - *>/ (1+ ") + G» + *)(1 - ") 



Sw 



4a sm — 
2 log — - + X - v] (1 .+ v) 

ITU 



(M+*)d -f) 



(167) 



acting at the center of the loaded area (Fig. 78). Expressions (165) and (167) are also 
applicable to the calculation of moments of a simply supported infinite strip as a 
particular case. 

1 It is permissible to regard them as concentrated provided u and v are small. 



SIMPLY SUPPORTED RECTANGULAR PLATES 



161 



Table 26. Values of the Factor <p and ^ Defined by Eqs. (163) 

A: = v/u 



k 


<P 


* 


k 


<p 


+ 


k 


<e 


* 





1.000 


-1.000 


1.0 


1.571 


0.000 


2.5 


1A27 


0.475 


0.05 


1.075 


-0.923 


1.1 


1.569 


0.054 


3.0 


1.382 


0.549 


0.1 


1.144 


-0.850 


1.2 


1.564 


0.104 


4.0 


1.311 


0.648 


0.2 


1.262 


-0.712 


1.3 


1.556 


0.148 


5.0 


1.262 


0.712 


0.3 


1.355 


-0.588 


1.4 


1.547 


0.189 


6.0 


1.225 


0.757 


0.4 


1.427 


-0.475 


1.5 


1.537 


0.227 


7.0 


1.197 


0.789 


0.5 


1.481 


-0.374 


1.6 


1.526 


0.261 


8.0 


1.176 


0.814 


0.6 


1.519 


-0.282 


1.7 


1.515 


0.293 


9.0 


1.158 


0.834 


0.7 


1.545 


-0.200 


1.8 


1.504 


0.322 


10 


1.144 


0.850 


0.8 


1.560 


-0.127 


1.9 


1.492 


0.349 


20 


1.075 


0.923 


0.9 


1.568 


-0.060 


2.0 


1.481 


0.374 


00 


1.000 


1.000 



Table 27. Values of the Factors X and /i (Eq. 166) for Simply Supported 

Rectangular Plates 







X for £/a 


= 






n for £/a 






b /a 
























0.1 


0.2 


0.3 


0.4 


0.5 


0.1 


0.2 


0.3 


0.4 


0.5 


0.5 


2.792 


2.352 


1.945 


1.686 


1.599 


0.557 


-0.179 


-0.647 


-0.852 


-0.906 


0.6 


2.861 


2.545 


2.227 


2.011 


1.936 


0.677 


0.053 


-0.439 


-0.701 


-0.779 


0.7 


2.904 


2.677 


2.433 


2.259 


2.198 


0.758 


0.240 


-0.229 


-0.514 


-0.605 


0.8 


2.933 


2.768 


2.584 


2.448 


2.399 


0.814 


0.391 


-0.031 


-0.310 


-0.404 


0.9 


2.952 


2.832 


2.694 


2.591 


2.553 


0.856 


0.456 


| 0.148 


-0.108 


-0.198 


1.0 


2.966 


2.879 


2.776 


2.698 


2.669 


0.887 


0.611 


0.304 


0.080 


0.000 


1.2 


2.982 


2.936 


2.880 


2.836 


2.820 


0.931 


0.756 


0.551 


0.393 


0.335 


1.4 


2.990 


2.966 


2.936 


2.912 


2.903 


0.958 


0.849 


0.719 


0.616 


0.578 


1.6 


2.995 


2.982 


2.966 


2.953 


2.948 


0.975 


0.908 


0.828 


0.764 


0.740 


1.8 


2.997 


2.990 


2.982 


2.975 


2.972 


0.985 


0.945 


0.897 


0.858 


0.843 


2.0 


2.999 


2.995 


2.990 


2.987 


2.985 


0.991 


0.968 


0.939 


0.915 


0.906 


3.0 


3.000 


3.000 


3.000 


2.999 


2.999 


0.999 


0.998 


0.996 


0.995 


0.994 


GO 


3.000 


3.000 


3.000 


3.000 


3.000 


1.000 


1.000 


1.000 


1.000 


1.000 



102 



THEORY OF PLATES AND SHELLS 



Extending the integration over circular, elliptic, and other areas, the corresponding 
expressions for M and N for these loadings are readily found. Taking, for instance, 
a circular loaded area (Fig. 79) we obtain for its center 



! 






i 
.o|c\J 


<~- £— > 






1 

1 

t ° 








1 

1 




$ 




jd|cvj 
1 








1 

* 








< a ■ 


> 





2a sin 



*S 



M =- 

4ir 
P 

N =- M 

4ir 



2 log 



irC 



+ X 



-) 



(168) 



these expressions being equivalent to the result (157). 
Comparing (168) with expressions (165) for ft = 1, we 
may conclude that a circular and a square loaded area 
are equivalent with respect to the bending moments they 
produce at the center of the area, if 



Fig. 79 



u 



c = 



V~2 



- e*' 4 - 1 = 0.57u or u = 0.88 X 2c (c) 



It should be noted that, as the load becomes more and more concentrated, the 
accuracy of the approximate logarithmic formulas for the bending moments, such as 
given by Eqs. (157) and (167), increases while the convergence of the customary series 
representing these moments becomes slower. Numerical calculations 1 show also that 
the accuracy of those approximate formulas is entirely sufficient for practical purposes. 

38. Thermal Stresses in Simply Supported Rectangular Plates, Let 

us assume that the upper surface of a rectangular plate is kept at a higher 
temperature than the lower surface so that the plate has a tendency to 
bend convexly upward because of nonuniform heating. Because of the 
constraint along the simply supported edges of the plate, which prevents 
the edges from leaving the plane of the supports, the nonuniform heat- 
ing of the plate produces certain reactions along the boundary of the 
plate and certain bending stresses at a distance from the edges. The 
method described in Art. 24 will be used in calculating these stresses. 2 
We assume first that the edges of the plate are clamped. In such a case 
the nonuniform heating produces uniformly distributed bending moments 
along the boundary whose magnitude is (see page 50) 



M n = 



atPjl + v) 
h 



(a) 



where t is the difference between the temperatures of the upper and the 
lower surfaces of the plate and a is the coefficient of thermal expansion, 

1 See S. Woinowsky-Krie^er, Ingr.-Arch., vol. 3, p. 340, 1932; and Ingr.-Arch., vol. 
21, pp. 336, 337, 1953. 

2 See paper by J. L. Maulbetsch, J. AppL Mechanics, vol. 2, p. 141, 1935; see also 
E. Melan and H. Parkus, " Warmespannungen infolge stationarer Temperaturf elder/ ' 
Vienna, 1953, which includes a bibliography on thermal stresses. For stresses due to 
assemblage errors in plates, see W. Nowacki, Bull. acad. polon. sci., vol. 4, p. 79, 1956. 



SIMPLY SUPPORTED RECTANGULAR PLATES 163 

To get the bending moments M x and M y for a simply supported plate 
(Fig. 62), we must superpose on the uniformly distributed moments given 
by Eq. (a) the moments that are produced in a simply supported rec- 
tangular plate by the moments M' n = -atD(l + v)/h uniformly dis- 
tributed along the edges. We shall use Eqs. (120) (see page 92) in dis- 
cussing this latter problem. Since the curvature in the direction of an 
edge is zero in the case of simply supported edges, we have M' t = vM' n . 
Hence at the boundary 

v _ M * ± M « = M « + M 'i = _ gggg ± v) (b) 

M ~ 1 + v 1 + v h 

Thus the first of equations (120) is satisfied by taking M constant along 
the entire plate and equal to its boundary value (b). Then the second 
of equations (120) gives 

d 2 w d 2 w _ at(l + v) / c ) 

dx 2 * dy 2 h 

Hence the deflection surface of the plate produced by nonuniform heat- 
ing is the same as that of a uniformly stretched and uniformly loaded 
rectangular membrane and is obtained by finding the solution of Eq. (c) 
that satisfies the condition that w = at the boundary. 

Proceeding as before, we take the deflection surface of the plate in the 
form ,« 

W = Wi + Wi \ a ) 

in which Wi is the deflection of a perfectly flexible string loaded uni- 
formly and stretched axially in such a way that the intensity of the load 
divided by the axial force is equal to -at{l + v)/h. In such a case the 
deflection curve is a parabola which can be represented by a trigonometric 

series as follows: 

at(\ + v) x(a - x) 
w,= 1 2— 

rrnrx 



sm 



at(l + v) 4a 2 V 2- (e) 

h tt 3 Aw 3 



m = 1,3,5, 



This expression satisfies Eq. (c). The deflection w 2 , which must satisfy 
the equation 

^_ 2 + d ^l = (/) 

dx 2 ^ dy 2 

can be taken in the form of the series 

oo 

~ (g) 



w,= 2j 


Y m sm 

a 


m — 1,3,5, . 





164 THEORY OF PLATES AND SHELLS 

in which Y m is a function of y only. Substituting (g) in Eq. (/), we find 



ra 2 ir 2 



Y" - '^4- Y m = 



a 2 



Hence Y m = A m sinh ^ + B m cosh ^ (*) 

From the symmetry of the deflection surface with respect to the x axis 
it may be concluded that Y m must be an even function of y. Hence the 
constant A m in the expression (h) must be taken equal to zero, and we 
finally obtain 

«-«* + «,- 2j sm -r[ h — » + Bm cosh a J 

wi = 113,5, • • • 

(0 

This expression satisfies the boundary conditions w = at the edges 
x = and a; = a. To satisfy the same condition at the edges y = ±6/2, 
we must have 

Bm C o S h — T — -j ^ " 

Substituting the value of B m obtained from this equation in Eq. (i), 
we find that 

<rf(l + v)4a 2 \ A I!l_fLli a ) (/) 



ir z h 



, v^\ sin / cosh—— A 

/ j m 8 \ cosh a m J 

tn = 1,3,5, • • • 



in which, as before, a m = nnrb/2a. 

Having this expression for the deflections w, we can find the corre- 
sponding values of bending moments; and, combining them with the 
moments (a), we finally obtain 



atDQ. + v) (dhv d*w\ 

Mx ~ h \3a; 2 ^ dy*J 



. rmrx , miry 

JT , „ ox v— \ sin cosh 

^ 4:Dat(l - v 2 ) \ A a <*_ 

*"** irh / j m cosh a m 

m = 1,3,5, ... j /tl\ 

•o . rnvx , miry 

x: — ^ sm cosn ■ • 

= «t(l - v 2 )D 4Dcrf(l - v 2 ) y a «_ 

"~ h irh / j m cosh a m 

m= 1,3,5, ... 



SIMPLY SUPPORTED RECTANGULAR PLATES 165 

The sum of the series that appears in these expressions can be readily 
found if we put it in the following form : 



00 

2 



. rrnrx , rrnry 

sin cosh — - 

a a 



111 = 1,3,5, 



m cosh a„ 



00 

-X 

m= 1,3,5, . 



(: 



. rrnrx , rrnry m , n . rrnrxy 

sin cosh — - e m7ryla sm 

a a a 



\ m cosh a^ 



me Q 



00 



e m7ry/o . rrnrx /IN 

+ / sin (Z) 

7 rae am a 



?n= 1,3,5, 



The first series on the right-hand side of this equation converges rapidly, 
since cosh {rrnry /a) and cosh a m rapidly approach e m7ry,a and e am as m 
increases. The second series can be represented as follows. 1 



, . rrnrx . tx 

e mwy/a g m g m 

= - arctan 7 r- (ra) 



v 

/ j me am 2 

m= 1,3,5, . . . 



\2a a ) 



The bending moments M z and M y have their maximum values at the 
boundary. These values are 

It is seen that these moments are obtained by multiplying the value of 
M n in formula (a) by (1 — v). The same conclusion is reached if we 
observe that the moments M' n which were applied along the boundary 
produce in the perpendicular direction the moments 

atD{\ + v) 



M' t = vM' n = -v 



h 



which superposed on the moment (a) give the value (n) . 

39. The Effect of Transverse Shear Deformation on the Bending of 
Thin Plates. We have seen that the customary theory of thin elastic 
plates leads to a differential equation (103) of the fourth order for the 

1 See W. E. Byerly, "Elementary Treatise on Fourier Series and Spherical, Cylin- 
drical and Ellipsoidal Harmonies," p. 100, Boston, 1893. The result can be easily 
obtained by using the known series 

1 2x sin if . , x z . X s . , 

- arctan = x sm <p + -— sm 3<p + ■— sin d<p -r • • " 

2 1 - x 2 3 5 



166 



THEORY OF PLATES AND SHELLS 



Qdxdy 



deflection and, accordingly, to two boundary conditions which can and 
must be satisfied at each edge. For a plate of a finite thickness, how- 
ever, it appears more natural to require the fulfillment of three boundary 
conditions than of two. The formal reason for the impossibility of satis- 
fying more than two conditions by the customary theory has been the 
order of the basic equation of this theory; physically this reason lies in 
the fact that the distortion of the elements of the plate due to transverse 
forces such as Q (page 52), Q x , and Q y (page 79) has been neglected in 
establishing the relations between the stresses and the deflection of the 
plate. The disregard of the deformation due to the transverse stress 
component obviously is equivalent to the assumption of a shearing 
modulus G z = oo ; proceeding in this way we replace the actual material 
of the plate, supposed to be isotropic, by a hypothetic material of no 
perfect isotropy. Owing to the assumption G z = <x> the plate does not 
respond to a rotation of some couple applied at the cylindrical surface 
of the plate, if the vector of the couple coincides with the normal to this 

surface. This enables us to identify 
the variation dM xy /dy of twisting cou- 
ples due to horizontal shearing stresses 

x' and acting along an edge x = a with 

the effect of vertical forces Q x applied 
at the same edge, thus reducing the 
jtiz i number of the edge conditions from 

three to two (page 83). The stress 
analysis of the elastic plates is greatly 
simplified by this reduction. On the 
other hand, in attributing some purely 
hypothetic properties to the material of the plate we cannot expect com- 
plete agreement of the theoretical stress distribution with the actual one. 
The inaccuracy of the customary thin-plate theory becomes of practical 
interest in the edge zones of plates and around holes that have a diameter 
which is not large in comparison with the thickness of the plate. 

The generalization of the customary theory with respect to the effect 
of shear deformation is substantially due to E. Reissner. 1 

Let us consider an element of the plate submitted to the external 
transversal load q dx dy and to a system of stress components (Fig. 80). 
In accordance with E. Reissner's theory we assilme a linear law for the 
distribution of the stress components a X} a y , and r xy through the thickness 
of the plate. By equations of equilibrium (a) on page 98 the distribu- 

1 See /. Math. andPhys., vol. 23, p. 184, 1944; J. AppL Mechanics, vol. 12, p. A-68, 
1945; Quart. AppL Math. } vol. 5, p. 55, 1947. For the history of this question going 
back to a controversy between M. Levy and Boussinesq, see L. Bolle, Bull. tech. Suisse 
romande, October, 1947. 




dy 






Fig. 80 



SIMPLY SUPPORTED RECTANGULAR PLATES 167 

tion of the components r xt and r yz then follows a parabolic law. As for 
the stress component a z it is readily obtained from the third of equations 
of equilibrium (a) if one takes into account the conditions 

(<r z )z—hit - — Q {oz)z=hn = 

at the upper and lower surface of the plate. We arrive, in this manner, 
at the following expressions for the stress components in terms of their 
resultants and the coordinate z : 



\2M x z _ l2M v z _ _ 12M xv z 

3Q.I\ (2z\-\ _3Q ;/ r (2z\-\ 



(a) 



•--?[i-¥+Kx)1 

Except for Eq. (6) the foregoing system of equations coincides with the 
corresponding relations of the customary theory. In like manner we can 
rewrite the following conditions of equilibrium of the stress resultants 
(see pages 80, 81) : 

!>Q* + *Qy + q = (c) 

dx ^ dy * 
dM x __ dM Xf , _ q = o 
dx dy x {d) 



dMy dM X y 



Qy = 



dy dx 

Assuming an isotropic material and supposing the displacements uq, v , 
Wo of any point of the plate to be small as compared with its thickness h, 
we make use of the general stress-strain relations 

— ° = -^ [<r x ~ A°u + *z)] 

OX lii 

-^ = ™ Wv - K** + **)] 
dy Jb 

duo dt>o 1 r p \ 

duo dWo __ 1 

~di + ~te ~ G Txz 

dv dWo __ 1 

~te + ~W ~~ G Tyz 
in which G = 2£/2(l + v). We do not use the sixth relation 



dWO 1 r / ! M 



168 THEORY OF PLATES AND SHELLS 

however, since this latter proves to be in contradiction with the assumed 
linear law for the distribution of the stresses <r x , a yi r xy . 

Next, 1 we introduce some average value w of the transverse displace- 
ment, taken over the thickness of the plate, as well as some average values 
<p x and (f y of the rotation of the sections x = constant and y = constant, 
respectively. We define these quantities by equating the work of the 
resultant couples on the average rotations and the work of the resultant 
forces on the average displacement to the work of the corresponding 
stresses on the actual displacements u , ^o, and w in the same section; 
i.e., we put 

J <r x uodz = M x <p x — / _ A/2 Wo dz = M xy <p v 

I CTyVodZ = My<Py — / T^yUo dZ = M X y<p X (/) 

f-h2 TxzWQdZ = ®* W f-h/2 TyzW ° dZ = ® yW 

Now, substituting expressions (a) for the stresses in Eqs. (/), we arrive at 
the following relations between the average and the actual displacements: 



12 p/ 2 UoZ , ,„s 

** = w]- h/2 -h dz i9) 

12 p/ 2 Vo2 , 

**=v }- h/2 T dz 

Using Eqs. (e) and observing Eq. (6), we are also able to express the 
stress components a x , <r v , and t xv in terms of the actual displacements; 
we find 2 



__ E /duo , dVo \ _ 
~ 1 - v*\dx + V dy) 

— % ( dv ° j_ du ° \ — 

~ 1 - v 2 \dt/ + V dx) 4(1 - v) 



4(1 - v) 
Sqv 



2 

3 

2 2z 

3 



2z 1 /2zV] 
h + Z\h) J 

M(!)1 



(h) 



Txv "" 2(1 + v) \ 



E /duo dVo\ 

dy + dx) 



1 E. Reissner, in his treatment of the subject, makes use of Castigliano's principle of 
least work to introduce the conditions of compatibility in the analysis. The method 
here followed and leading to substantially the same results is due to A. E. Green, 
Quart. Appl. Math., vol. 7, p. 223, 1949. See also M. Schafer, Z. angew. Math. Mech., 
vol. 32, p. 161, 1952. V 

2 Terms with z 3 do not actually occur in the following expressions for a x and o> 
since they are canceled out by identical terms with opposite sign contained in du /da 
and dvo/dy. 



SIMPLY SUPPORTED RECTANGULAR PLATES 169 

Substituting this in Eqs. (a), multiplying the obtained equations by 
\2zdz/h z , integrating between z = — h/2 and z = h/2, and observing 
relations (g), we arrive at the expressions 



dy ' 5Eh 



— v) (d<p x d<p v \ 
> \dy ^ dx) 



(0 



in which D is denned, as before, by Eq. (3). In like manner, substituting 
expressions (a) for the stress components r xz and r yz in the last two equa- 
tions (e), multiplying the result by f[l — (2z/h) 2 ] dz/h, and integrating 
between the limits z = ± h/2, we obtain 

dw , 12 1 + v n 
** ~ ~ dx~ + T ~Eh~ Qx 

dw , 12 1 + v n {J) 

«» = ~-dy- + T~Eh Qv 

Now, eight unknown quantities, namely M X9 M yj M xyy Q x , Q y , w, <p x , and 
ipy 9 are connected by two equations (j), three equations (i) } and, finally, 
by three equations of equilibrium (c) and (d). 

In order to transform this set of equations into a form more convenient 
for analysis we eliminate the quantities <p x and <p y from Eqs. (j) and (i), 
and, taking into account Eq. (c), we obtain 



dQ x qh 



n/r t^ /3 2 w . d 2 w\ . h 2 

M * = - D {w +v w) + -5 dx 10 1 



h 2 dQ v qh 2 



■** n ( d 2 w , d 2 w\ , 

M v=- D (w+>>w)+ 5 dy 10 1 



(fc) 



Substitution of these expressions in Eqs. (d) yields, if one observes Eq. 
(c), the result 

V, 10 AVx ^ ax 10(1 - v) da: 

Vy 1Q A^ y ^ ^ 1Q(1 _ ^ ^ 

in which, as before, the symbol A has the meaning (105). In the par- 
ticular case of h = 0, that is, of an infinitely thin plate, the foregoing 
set of five equations, expressions (k) and (/), gives Eqs. (101) and (102) 



170 THEORY OF PLATES AND SHELLS 

for the moments and Eqs. (108) for the shearing forces of the customary 
thin-plate theory. 

To obtain the more complete differential equation for the deflection 
of the plate we only have to substitute expressions (I) in Eq. (c); thus 
we obtain 

DAAw = q - 5s V 12 A? <!«») 

* 10 1 — v 

We can satisfy this equation by taking w, that is, the "average deflection" 
at {x,y), in the form 

w = w' + w" ( m ) 

in which w' is a particular solution of the equation 

t. h 2 2 - v . , * 

5 10 1 — v 

and w" is the general solution of the equation 

AAw" = (o) 

Therefore, using Eq. (169), we are able, just as in the ordinary thin-plate 
theory, to satisfy four boundary conditions in all. We can obtain a sup- 
plementary differential equation, however, by introducing into consider- 
ation the shearing forces Q x and Q y . Equation of equilibrium (c) is 
satisfied, in fact, if we express these forces in a form suggested by the 
form of Eqs. {I), i.e., 

Qx ~ «» + dy {) 

n - _n^) -*t 
^ y ~ dy dx 

or Q*- Q *- D —dlT+ dy () 

Vy ~ V» u dy dx 

In these expressions ^ denotes some new stress function, whereas Q' x and 
Q' y must satisfy the relations 

Q> h * ACT - - D ^^ ^— ^ . 

«* - 10 A ^ ~ U dx 10(1 - v) dx (r) 

Q y ~ 10 Wv " 32/ 10(1 - y) dy 

as we can conclude from Eqs. (I) and (n). Differentiating the foregoing 
equations with respect to x and y, respectively, and adding the results 



SIMPLY SUPPORTED RECTANGULAR PLATES 171 

wc arrive at the condition of equilibrium 

d 4'~ + ¥* + <? = (s) 

dx ay 

To establish a differential equation for the stress function ^ we substitute 
expressions (q) in Eqs. (/) with the result 



£(*-&*)--£('-M- 



(0 



from which we conclude that the expressions in parentheses are con- 
stants. Making these constants equal to zero we have the relation 

A ^_ ^ = (170) 

which, still assuming that h ^ 0, yields a second fundamental equation 
of the generalized theory of bending, in addition to Eq. (169). 

Having established two differential equations, one of which is of the 
fourth and the other of the second order, we now are able to satisfy three 
conditions, instead of only two, on the edge of the plate. Considering 
the general case of an element of the cylindrical boundary of the plate 
given by the directions of the normal n and the tangent t (Fig. 54) we can, 
for instance, fix the position of the element by the equations 



W = W <p n = <Pn <Pt — <Pt 



(u) 



Herein w is the given average deflection and <p n and <p t are the given 
average rotations of the element with respect to the axes t and n respec- 
tively. In the particular case of a built-in edge the conditions arej/; = 0, 
<p n = 0, and (ft = 0. Instead of displacements some values Q n , M n , 
M nt of the resultants may be prescribed on the boundary, and the corre- 
sponding edge conditions would be 

Qn = Qn M n = M n M nt = M n t (v) 

Hence the conditions along a free edge are expressed by equations Q n = 0, 
M n = 0, Mnt = 0, and for a simply supported edge the conditions are 
w = 0, M n = 0, M 7l t = 0. In the latter case we obtain no concentrated 
reactions at the corners of the plate, which act there according to the 
customary theory and are in obvious contradiction to the disregard of 
the shear deformation postulated by this theory. 

As an illustration of the refined theory let us consider a plate in form of a semi- 
infinite rectangle bounded by two parallel edges y = 0, y = a and the edge x = 0. 
We assume that there is no load acting on the plate, that the deflections iv and the 



172 THEORY OF PLATES AND SHELLS 

bending moments M v vanish along the edges y = 0, y = a, and that the edge % = 
is subjected to bending and twisting moments and to shearing forces given by 

Mz = Mo sin 

a 

M xy = Ho cos (w) 

a 

Qx = Qo sin — 
a 

where M o, # o, Qo are constants and n is an integer. Then, in view of q = 0, we have 
w r = by Eq. (n) and w = w" by Eq. (ra). We can satisfy Eq. (o) and the condition 
of vanishing deflections at x = oo by taking 

w = u?" = sin 



(nwx S\ e~ nTX 



a 
A and B being any constants. Next, assuming for ^ a solution of the form 

v niry 
ty = X cos 

a 

where X is a function of x alone, and substituting this in Eq. (170) we obtain 

\p = Ce~ x P cos 

a 

In this last expression 

ZnV 2 10 

and C is a constant. From Eqs. (r) we have Q^ = Q' y = and Eqs. (q) give 

Q* = - \2B ( — ) e-»™'« + C — «-** sin — - 

L \ a / a J a 

Finally, Eqs. (A;) yield the following expressions for moments acting along the edge 
x = 0: 

V 2 . n^ri/ 



sin 
a 



WW- - [ -^(1 - ,) + 2B ^1 + — j + C — J — 

<M„)„. - [ -Ml - ,) + B [l - , + j — j + <^— + ? jj — co, _ 
Equating these expressions, together with the expression for the shearing force 

«?*)*-o - - \2B I — ) + C— sin— £ 
L \ a / a J ° 

to the expressions (w), respectively, we obtain a set of three equations sufficient to 
calculate the unknown constants A, B, and C. In this way, by using the refined plate 
theory, all three conditions at the edge x = are satisfied. 



SIMPLY SUPPORTED RECTANGULAR PLATES 



173 



Considering now the edges y = we see that w vanishes along those edges, and M y 
aiso vanishes there, as can be proved by substituting the expression for Q y into 
the second of equations (k). 

Another theory of plates that takes into account the transversal shear deformation 
has been advanced by A. Kromm. 1 This theory neglects the transverse contraction 
e z but, in return, does not restrict the mode of distribution of bending stresses across 
the thickness of the plate to a linear law. Applying this theory to the case of a 
uniformly loaded, simply supported square plate with a/h = 20, Kromm found the 
distribution of shear forces acting along the edge as shown in Fig. 81. For comparison 
the results of customary theory (Fig. 63) are also shown by the dashed line and the 



0.5 qa 




v R = 0.065qa 2 
0.5qa 



1.0qa, 



1.5qa 



Fig. 81 



forces R. We see that, as soon as the transversal shear deformation is taken into 
account, no concentrated reaction is obtained at the corner point of the plate. The 
corresponding negative forces are distributed instead over a small portion of the 
boundary adjacent to the corner, yielding at the corner itself a finite pressure acting 
downward. The moments M xy on the four sides of the plate are zero in that solution. 

Still another approach to the theory of shear deformation can be found in a paper 
of H. Hencky. 2 

40. Rectangular Plates of Variable Thickness. 3 In deriving the differential equa- 
tion of equilibrium of plates of variable thickness, we assume that there is no abrupt 
variation in thickness so that the expressions for bending and twisting moments 
derived for plates of constant thickness apply with sufficient accuracy to this case also. 
Then 



\dx 2 dy 2 / \dy 2 dy 2 / 



M xy = -M vx = D(l - v) 



dhu 
dx dy 



(a) 



1 A. Kromm, Ingr.-Arch., vol. 21, p. 266, 1953; Z. angew. Math. Mech., vol. 35, p. 
231, 1955. 

2 Ingr.-Arch., vol. 16, p. 72, 1947. 

3 This problem was discussed by R. Gran Olsson, Ingr.-Arch., vol. 5, p. 363, 1934; 
see also E. Reissner, /. Math, and Phys., vol. 16, p. 43, 1937. 



174 



THEORY OF PLATES AND SHELLS 



Substituting these expressions in the differential equation of equilibrium of an element 
[Eq. (100), page 81], 



d 2 M x aw iy f d 2 M v 
dx 2 dx dy dy 2 



(b) 



and observing that the flexural rigidity D is no longer a constant but a function of the 
coordinates x and ?/, we obtain 

dD d , rt dD d 

DAAw +2 Aw +2 Aw 

dx dx dy dy 



+ 



, (d 2 D dhv rt d 2 D dhv , d 2 D d 2 w\ 
ADAw - (1 - v) ^—--2— — + — -)= q (171) 

where, as before, we employ the notation 

d 2 d 2 

A = 1 

dx 2 dy 2 

As a particular example of the application of Eq. (171) let us consider the case in 
which the flexural rigidity D is a linear function of y expressed in the form 

(c) 



D = Do + D x y 



where D* and Di are constants. 




Fig. 82 



In such a case Eq. (171) becomes 



or 



(Do + Diy)AAw + 2Di — Aw = q 

dy 

A[(D + Diy) Aw] = q 



(172) 



Let us consider the case in which the intensity of the load q is proportional to the 
flexural rigidity D. We shall assume the deflection of the plate (Fig. 82) in the form 

W = W\ + 102 

and let w\ equal the deflection of a strip parallel to the x axis cut from the plate and 
loaded with a load of intensity 



-<"( i+ i->) 



(<0 



SIMPLY SUPPORTED RECTANGULAR PLATES 175 

This deflection can be represented, as before, by the trigonometric series 



(Do + D x y)ir b J /j m h * m a w b D / j m* 



W\ = ■ — /T , , ^ r ~. / — ; sm • — — = -777" / ~~7 sin ( 6 ) 



m= 1,3,5, ... m = 1,3,5, . 

By substitution we can readily show that this expression for w\ satisfies Eq. (172). 
It satisfies also the boundary conditions w\ = and d 2 wi/dx 2 = along the supported 
edges x = and x = a. 

The deflection w> 2 must then satisfy the homogeneous equation 

A[(7^ + D lV ) Aw 2 ] = (/) 

We take it in the form of a series 



oo 

I 



\ ^ • mirX i ^ 

w 2 = / J m sin (g) 



m= 1,3,5, . 



Substituting this series in Eq. (/), we find that the functions Y m satisfy the following 
ordinary differential equation : 



Using the notation 



/ „ ??i 2 7r 2 m \ mir 

u = (Do + d iV ) ( f;: — r Y m ) — = 

y a 1 J a 



(0 



we find, from Eq. (ft), 

Then, from Eq. (?*), we obtain 

v" 2 v A »** mV +B m e- a m v 

Do + D\\J 

The general solution of this equation is 

Ym = C m e<*mv + D m e-«>nv + ^ (jfc) 

in which g m is a particular integral of Eq. (j). To find this particular integral we use 
the Lagrange method of variation of constants; i.e., we assume that g m has the form 

g m = E m e a >»v + F vl e- a »<y (I) 

in which E m and F m are functions of y. These functions have to be determined from 
the following equations: 1 

E' m **** + F' m e-«mv = o 

A m e«my + B m e~«mv 

J$ e a m y _ p e -a m V = . 

a m (D +D iy ) 
1 E* m and F' m in these equations are the derivatives with respect to y of E m and F m . 



176 THEORY OF PLATES AND SHELLS 

from which 

A m + Bme-^rnV 



F* = - 



2« m (Z> + D lV ) 

AneH'mV + B T 



2a m (Z> + D iV ) 
Integrating these equations, we find 

E m - / - " .T. - - , dy = — — log — (Do + Dm) 
„(D + Diy) 2a n Di Di 

-2am(Dt+Diy) 



B m ^ J e J ' d[2«,(P. j- JP,y)] 

+ 2« m D, 6 ' J 2a m (£> + D lV )^ 

A m e ia «>v + B m B m 2<x m 

dy = — log -—— (Do + Diy) 

2a m (Do + D x y) 2a m D 1 6 Di 

2a„(£>0 + i)iv) 

*>* d[2a m (D„ + Dm)] 



[ A^ 
J 2a„ 

2cr m Z)o r 

F..-J 

- C &l / 

2a m Di J 2a m (D + Diy) 

Substituting these expressions in Eqs. (/) and (k) and using the notation 1 

/u e u f u e~ u 

-du Ei(-u) - / — du 
-co U J* U 

we represent functions Y m in the following form: 

F m = A' m llog -^ (Do + D lV ) - 6 * Ei [ — J J *■■* 

. j ^(Do + Dn/) , 2<*„, , n . _ r -2a w (Do + i)iT/) 11 
- 5 TO U ^ log — (Do + Di») - Ei I — J j e a mv 

+ Cme"™* + D m e- a ^ (m) 

The four constants of integration A' m , B' m , C m , D m are obtained from the boundary 
conditions along the sides y = and y = b. In the case of simply supported edges 
these are 

The numerical results for a simply supported square plate obtained by taking only the 
first two terms of the series (g) are shown in Fig. 83. 2 The deflections and the 
moments M x and M y along the line x = a/2 for the plate of variable thickness are 
shown by full lines; the same quantities calculated for a plate of constant flexural 
rigidity D = %(D + Dib) are shown by dashed lines. It was assumed in the calcula- 
tion that DJ> = 7D and v = 0.16. 

1 The integral E%(u) is the so-called exponential integral and is a tabulated function; 
see, for instance, Jahnke-Emde, " Tables of Functions/' 4th ed., pp. 1 and 6, Dover 
Publications, 1945; or "Tables of Sine, Cosine and Exponential Integrals/' National 
Bureau of Standards, New York, 1940. 

1 These results are taken from R. Gran Olsson, loc. cit. 



SIMPLY SUPPORTED RECTANGULAR PLATES 



177 



Finally, let us consider the case in which the thickness of the plate is a linear function 
of y alone and the intensity of the load is any function of y (Fig. 82). Denoting the 
thickness of the plate along the line y = b/2 by h and the corresponding fiexural 
rigidity by 

Ehl 



Do = 



12(1 - v*) 



we have at any point of the plate 

/i 3 



D = Do 



and 



*-[i + x(H; 



(n) 



(o) 



where X is some constant. This yields h = (1 . — \)h at y = and h = (1 -}- X)ft at 
y =6. 

M • 4q0 ° 2 




y = 0.8/2 



y = 0.972 



MomenT My Moment M* Deflection 
Fig. 83 

The following method 1 introducing the quantity X as a parameter proves to be most 
efficient in handling the present problem. Considering the deflection w as a function 
of the variables x } y, and X, we can express w(x,y,\) in form of the power series 



w 



m = 



(p) 



in which vi is an integer and the coefficients w; 7n are merely functions of x and y. 
'See II. Favre and B. Gilg, Z. angew. Math. u. Phys., vol. 3, p. 354, 1952. 



X78 THEORY OF PLATES AND SHELLS 

Substituting expressions (o) and (p) in Eq. (171) and equating to zero the coeffi- 
cients of successive powers of X, we obtain a sequence of differential equations 

AAwo = — 
■Do 



'-- 3 Di a "* + (?~') i4 "'°] 

•--'[i5* ,,+ (t ->) 4 H 
f 8 r x ^oi 



(«) 



We assume the edges x = and a; = a to be simply supported, and we shall restrict 
the problem to the case of a hydrostatic load 

q =— (r) 

Using the method of M. Levy we take the solution of Eqs. (q) in the form 

W 
(0 
(«) 



the coefficients F mn (rre = 0, 1, 2, . . .) being some functions of y. We can, finally, 
represent the load (r) in analogous manner by putting 



wo= If 


. n7T£ 

F on sin 

a 


n-1,3, 




00 




».- 2) 


I m sm 

a 


t? = 1,3, 




00 




wm= X 


y m „ sm 

a 


n = l,3, 





00 f 

fyoy X} l . ™*-z 

- — - > - sm 

™ / ^ n a 



n a 

n=l,3, ... 

Substitution of expressions (s) and (v) in the first of the equations (q) enables us to 
determine the functions Y on) the boundary conditions being Y on = 0, Y on = at 
y = and y = b if these edges are simply supported. The substitution of expressions 
(s) and (t) in the second of the equations (q) yields the function Yin. In like manner 
any function w m is found by substitution of wo, Wi, . . . , w m -i in that differential 
equation of the system (g) which contains w m at the left-hand side. The procedure 
remains substantially the same if the edges y = 0, b are built-in or free instead of being 
simply supported. 



SIMPLY SUPPORTED RECTANGULAR PLATES 



179 




M x :qoQ 2 ^- 
0.03 



0.25 



0.50 



0.75 



1.00 




Moment M x 



Fig. 84 



0.25 



0.50 



0.75 



1.00 



Numerical results obtained by H. Favre and B. Gilg 1 for the deflections and the 
I M-nding moments along the center line x = a/2 of a simply supported plate with 
X = 0.2 and v = 0.25 under hydrostatic pressure (r) are shown in Fig. 84. Full lines 
«!jvo results obtained by taking three terms in the series (p), while the dashed lines 
hold for the result of the first approximations. 

1 Ibid. 



CHAPTER 6 

RECTANGULAR PLATES WITH VARIOUS 
EDGE CONDITIONS 



41. Bending of Rectangular Plates by Moments Distributed along the 
Edges. Let us consider a rectangular plate supported along the edges 
and bent by moments distributed along the edges y = ±6/2 (Fig. 85). 
The deflections w must satisfy the homogeneous differential equation 

dx* ^ dx* dy* ^ dy* w 



and the following boundary conditions 

dx* 



w = o ?-^ = for x = and x = a (6) 




w = for y = + ^ ( c ) 

\dy i /y = b/2 \dy 2 / v =-b/2 

. . in which /i and / 2 represent the bending 

/-y- r ^~ r* /7 moment distributions along the edges 

y'= ±b/2. 

We take the solution of Eq. (a) in the 




U M yAfci, form of the series 

/r r r r r c 

jk- a V 



W 



Fig. 85 



= > 7 m sm — (e) 



each term of which satisfies the boundary conditions (b). The functions 
F m we take, as before, in the form 

y. = ^ sinh ZSL + B m cosh =3 + ft. =? sinh =8 



a a 



which satisfies Eq. (a). 

180 



VARIOUS RECTANGULAR PLATES 181 

To simplify the discussion let us begin with the two particular cases: 

1. The symmetrical casein which (M y ) y==b/2 = (M y ) ys= - b/2 

2. The antisymmetrical case in which (M y ) y==b/2 = — (M y ) ys =-. b/2 

r rhe general case can be obtained by combining these two particular cases. 

In the case of symmetry Y m must be an even function of y, and it is 

necessary to put A m = D m = in expression (/). Then we obtain, from 

00 

w = V ( B m cosh ^y + Cm ^M sinh W) sin nz* ( } 

To satisfy the boundary condition (c) we must put 

B m cosh a m + CmOi m sinh a m = 

where, as before, 

rrnrb 

Hence B m = —C m a m tanh a m 

:i.nd the deflection in the symmetrical case is 

00 

X~ (miry . , rrnry , ra7ry\ . wi7r;r /7N 
C m ( sinh — a m tanh a m cosh ) sin (h) 
\ a a a J a 

m—l 

We use the boundary conditions (d) to determine the constants C m . 
I {(^presenting the distribution of bending moments along the edges 
// ■--= ±6/2 by a trigonometric series, we have in the case of symmetry 



00 

fi(x) = f 2 (x) = / E m sin ^^ 

ra = 1 



(*") 



where the coefficients E m can be calculated in the usual way for each 
Particular case. For instance, in the case of a uniform distribution of 
1 1 in bending moments we have (see page 151) 

00 

m= 1,3,5, .. . 

Substituting expressions (h) and (i) into conditions (d), we obtain 

00 00 

rt r> V m27r " 2 /^ u • wtto; V^ _, . m7ra: 

-•*/> / — r- C m cosh a m sin = / E m sin 

Zv or a Z^ a 



?n = 1 



182 THEORY OF PLATES AND SHELLS 

from which 

a*E„ 

Cm — 



2Dw 2 tt 2 cosh a* 

and 

rmrx 



M . II Wl •*/ 

- _^_ V S 2 m y E m (a m tanh «. cosh *«* - *& sin h ^) 
2x 2 D/L/ m 2 cosha m m \ a a a/ 



w 

TO=1 



(173) 

In the particular case of uniformly distributed moments of intensity M 
we obtain, by using expression (j) , 

.2*-.' V __» («. tanh a. cosh 5S 

ic d D Z-/ wi 3 cosh a m \ a 



w 

m= 1,3,5 



ra7r# . , W7ri/\ . mx# 

smh — - ) sin 

a a / a 

The deflection along the axis of symmetry (y = 0) is 

00 

(")— -"Tin" 2/ m' cosher blU_ T W 

m- 1,3,5, ... 

When a is very large in comparison with 6, we can put tanh a m ~ <x m 
and cosh a m « 1. Then, by using series (j), we obtain 

00 

, x M & 2 V 1 • mirX * ^o 62 



?n= 1,3,5, . . . 



This is the deflection at the middle of a strip of length b bent by two 
equal and opposite couples applied at the ends. 

When a is small in comparison with b, cosh a m is a large number, and 
the deflection of the plate along the x axis is very small. 

For any given ratio between the lengths of the sides of the rectangle 
the deflection at the center of the plate, from expression (/c), is 

oo 

_ M «a6 V , n(m _ 1)/2 J_tanh«!? 
(w)«-o.*-«/2 - ^tq Lf y ' m 2 cosh a m 

m= 1,3,5, . . . 

Having expression (173) for deflections, we can obtain the slope of the 
deflection surface at the boundary by differentiation, and we can calcu- 
late the bending moments by forming the second derivatives of w. 



VARIOUS RECTANGULAR PLATES 



183 



Some values of the deflections and the bending moments computed in 
this way are given in Table 28. It is seen, for example, that the deflec- 
tion of a strip of a width a is about 3^ times that of a square plate of 
dimensions a. While the transverse section at the middle of a strip 
transmits the entire moment Mo applied at the ends, the bending moment 
My at the center of the plate decreases rapidly as compared with M , 
with an increasing ratio b/a. This is due to a damping effect of the edges 
.*; = and x = a not exposed to couples. 

Table 28. Deflections and Bending Moments at the Center of 

Rectangular Plates Simply Supported and Subjected to 

Couples Uniformly Distributed along the Edges y = ±6/2 (Fig. 85) 

v = 0.3 



b/a 


w 


M x 


My 





0.1250M 6 2 /£> 


0.300Mo 


l.OOOMo 


0.50 


0.0964M„b 2 /Z> 


0.387M« 


0.770Mo 


0.75 


0.0620Afo&7-D 


0.424M 


0.476M 


1.00 


0.0368Af a 2 AD 


0.394M 


0.256Mo 


1.50 


0.0280M a7Z> 


0.264M 


0.046M 


2.00 


0.0174M aV.D 


0.153M 


-O.OlOMo 



Let us consider now the antisymmetrical case in which 

j- / \ j- i \ \^ n • m7rX 

fi(x) = —J2(x) - / E m sin 



m= 1 



lii this case the deflection surface is an odd function of y, and we must 
put B m = C m = in expression (/). Hence, 



w 



QO 

V (A m sinh ™« + Dm ™« cosh ^l) 
Lj \ a a a ) 

m — 1 



sin 



rrnrx 



a 



roni the boundary conditions (c) it follows that 

A m sinh a m + D m a m cosh a m = 



WlHMiCe 
:ilifl 

W 



D m = tanh a m A, 

OL m 



oo 

Z 4 / . . miry 1 , . miry , nnry\ . rrnrx 
A m \ smh — tanh a m cosh 1 sin 
\ a a m a a } a 

m=} 



184 THEORY OF PLATES AND SHELLS 

The constants A m are obtained from conditions (d), from which it follows 
that 

2t 2 D V a w 2 . , , , . rmrx V t? • m7rX 

— — > A m — sinh a m tanh a m sin = / h m sin — — 

a 2 jL/ oc m a Lj a 

m=l m=l 

Hence A m = ^q #* m 2 sinh «I tanh <* m 

and 

= T7n / 2 'u I «~ coth a - smh "^T 7T COsh n ) 

27r 2 D Z-/ m 2 sinh a m \ a a a / 



m— 1 



sin — (174) 
a 



We can obtain the deflection surface for the general case represented 
by the boundary conditions (d) from solutions (173) and (174) for the 
symmetrical and the antisymmetrical cases. For this purpose we split 
the given moment distributions into a symmetrical moment distribution 
My and an antisymmetrical distribution M' y ', as follows: 

These moments can be represented, as before, by the trigonometric series 

00 

r y)v=bl* = X 

00 



(M:) y=6/2 = )E' m sin 



m-1 (J) 



C sin 

m a 



m=l 



and the total deflection is obtained by using expressions (173) and (174) 
and superposing the deflections produced by each of the two foregoing 
moment distributions {I). Hence 

* . mirx 

w — 



TTTn / r~ — >f— ( a m tanh a m cosh — 

27r 2 D / ^ m 2 Lcosh a m \ a 

"V S inh rm) + *^L- fa. coth a m sinh ^ 
a a / smh a m \ a 



w=l 

miry 



^osh^lj (175) 



^1 



VARIOUS RECTANGULAR PLATES 



185 



If the bending moments M y = \ E m sin (rrnrx/T) are distributed only 



771 = 1 



along the edge y = 6/2, we have / 2 (x) = 0, E' m = E^ = p/ m ; and the 
deflection in this case becomes 



w = 



7n— 1 

?n7r?/ . , m7ri/\ . 1 / . . . 7n?ri/ 

sinh J + -v-r ( «m coth a m smh — - 

a a / sinh a m \ a 



-^cosh^Yj (176) 
a a /J 

Solutions (173) to (176) of this article will be applied in the investigation 
of plates with various edge conditions. 

Moments Mo distributed along only one edge, say y = 6/2, would pro- 
duce, at the center of the plate, one-half the deflections and bending 
moments given in Table 28. In case of a simultaneous action of couples 
along the entire boundary of the plate, the deflections and moments can 
be obtained by suitable superposition of the results obtained above for a 
partial loading. 1 

42. Rectangular Plates with Two Opposite Edges Simply Supported 
and the Other Two Edges Clamped. Assume that the edges x = and 
x = a of the rectangular plate, shown in Fig. 86, 
are simply supported and that the other two edges 
are clamped. The deflection of the plate under 
any lateral load can be obtained by first solving 
the problem on the assumption that all edges are 
simply supported and then applying bending 
moments along the edges y = ±6/2 of such a 
magnitude as to eliminate the rotations produced 
along these edges by the action of the lateral load. 
In this manner many problems can be solved by 
combining the solutions given in Chap. 5 with the 
solution of the preceding article. 

Uniformly Loaded Plates. 2 Assuming that the edges of the plate are 
simply supported, the deflection is [see Eq. (139), page 116] 

1 Bending by edge couples was also discussed by H. Bay, Ingr.-Arch., vol. 8, p. 4, 
1937, and by U. Wegner, Z. angew. Math. Mech., vol. 36, p. 340, 1956. 

2 Extensive numerical data regarding rectangular plates with uniform load and sides 
simply supported or clamped in any combination may be found in a paper by F. 
Czerny; see Bautech.-Arch., vol. 11, p. 33, Berlin, 1955. 



'////////////// 



'////////////// 



L o +, 



T 
i 

-OJCVJ 



-*- 



ja|cvj 



Fig. 86 



186 THEORY OF PLATES AND SHELLS 

— 4ga 4 V J_ o . rmrx A _ a m tanh a m + 2 ^ miry 
~~ t 5 D Z/ m 6 jm a \ 2 cosh a m a 



it? 

m= 1,3,5, . 



+ _*_ H2L sinh 2E*\ (a) 



2 cosh a m a 
and the slope of the deflection surface along the edge y = 6/2 is 

00 

/dw\ = 2^ V 1 gin rmnr _ tanh am(1 + ^ tanh ffm)] 

171== 1,3,5, . . . 

To eliminate this slope and thus to satisfy the actual boundary conditions 
we distribute along the edges y = ±b/2 the bending moments M v given 
by the series 

eo 

{M y ) y =± m = 2^ E m sin ^ (c) 



TO= 1 



and we determine the coefficients E m so as to make the slope produced 
by these moments equal and opposite to that given by expression (6). 
Using expression (173) x for the deflection produced by the moments, we 
find that the corresponding slope along the edge y = b/2 is 



oo 

WD jLf 



. rrnrx 
sin 



— 2£ m [tanh <x m (a m tanh a m — 1) — a m ] (d) 



2ttD j/^ m 

m — 1,3,5, •• . 

Equating the negative of this quantity to expression (6), we find that 

jp - 4ga 2 a m — tanh a m (l + a m tanh a m ) , x 

m ~~ 7r 3 m 3 a m — tanh a m (a m tanh a m — 1) 

Hence the bending moments along the built-in edges are 

00 , . rrnrx 
_ 4ga 2 >y Sm a a m - tanh a m (l + a m tanh a m ) (/) 
(My) ys=±h /2 - — r £j m 3 am _ tan h am ( am tanh a m - 1) 

m — 1,3,5, • • • 

The maximum numerical value of this moment occurs at the middle of 
the sides, where x = a/2. Series (/) converges rapidly, and the maxi- 
mum moment can be readily calculated in each particular case. For 

1 From the symmetry of the deflection surface produced by the uniform load it can 
be concluded that only odd numbers 1, 3, 5, . . . must be taken for m in expression 
(173). 



VARIOUS RECTANGULAR PLATES 



187 



example, the first three terms of series (/) give — 0.070ga 2 as the maxi- 
mum moment in a square plate. In the general case this moment can be 
represented by the formula yqa 2 , where 7 is a numerical factor the magni- 
tude of which depends on the ratio a/b of the sides of the plate. Several 
values of this coefficient are given in Table 29. 

Substituting the values (e) of the coefficients E m in expression (173), 
we obtain the deflection surface produced by the moments M y distributed 

Table 29. Constants a, /3i, /3 2 , 7 for a Rectangular Plate with Two 
Edges Simply Supported and Two Edges Clamped (Fig. 86) 

v = 0.3 
b < a 



a 


a n 

* = 2'- y = ° 

qb* 


x = 2'- y = ° 


a n 

x = 2' y = ° 


a b 

X " 2' y - 2 


b 


M* = frqb* 


My = foqb 2 


ilf , = 7g& 2 




a 


Pi 


ft 


7 


00 


0.00260 


0.0125 


0.0417 


-0.0833 


2 


0.00260 


0.0142 


0.0420 


-0.0842 


1.5 


0.00247 


0.0179 


0.0406 


-0.0822 


1.4 


0.00240 


0.0192 


0.0399 


-0.0810 


1.3 


0.00234 


0.0203 


0.0388 


-0.0794 


1.2 


0.00223 


0.0215 


0.0375 


-0.0771 


1.1 


0.00209 


0.0230 


0.0355 


-0.0739 



b > a 



b 
a 


qa* 

Wm»j = a -— - 


x = 2' y = 

Mx = /3,<za 2 


x = -> y = 
My = fcqa 2 


a b 

X = -) v = — 
2 y 2 

My = 7^a 2 




a 


ft 


Pi 


7 


1 


0.00192 


0.0244 


0.0332 


-0.0697 


1.1 


0.00251 


0.0307 


0.0371 


-0.0787 


1.2 


0.00319 


0.0376 


0.0400 


-0.0868 


1.3 


0.00388 


0.0446 


0.0426 


-0.0938 


1.4 


0.00460 


0.0514 


0.0448 


-0.0998 


1.5 


0.00531 


0.0585 


0.0460 


-0.1049 


1.6 


0.00603 


0.0650 


0.0469 


-0.1090 


1.7 


0.00668 


0.0712 


0.0475 


-0.1122 


1.8 


0.00732 


0.0768 


0.0477 


-0.1152 


1.9 


0.00790 


0.0821 


0.0476 


-0.1174 


2.0 


0.00844 


0.0869 


0.0474 


-0.1191 


3.0 


0.01168 


0.1144 


0.0419 


-0.1246 


00 


0.01302 


0.1250 


0.0375 


-0.1250 



Igg THEORY OF PLATES AND SHELLS 

along the edges 



. mwx 
sin 



2qa A \ a 

Wl ** tt*D / / m h cosh a m 

m= 1,3,5, ... 

a m - tanh a m {\ + am tanh a m ) f rrnry ^^ mwy 
a m — tanh a m (a m tanh a m — 1) \ a a 



— a m tanh a m cosh — — J (g) 



miry \ 
a J 



The deflection at the center is obtained by substituting x = a/2, y = 
in expression (gr). Then 

00 

2ga 4 V (-1) (OT ~ 1)/2 am tanh a ro 

(W>l)max - ^ftjp) ^ m 6 cos h « m 

a m — tanh a OT (l + a m tanh a m ) 
a m — tanh a m (a m tanh a m — 1) 

This is a rapidly converging series, and the deflection can be obtained 
with a high degree of accuracy by taking only a few terms. In the case 
of a square plate, for example, the first term alone gives the deflection 
correct to three significant figures, and we obtain 

Wi = 0.00214 ^L 

Subtracting this deflection from the deflection produced at the center by 
the uniform load (Table 8, page 120), we obtain finally for the deflection 
of a uniformly loaded square plate with two simply supported and two 
clamped edges the value 

w = 0.00192 ^ 

In the general case the deflection at the center can be represented by the 

formula 

qa* 

Several values of the numerical factor a are given in Table 29. 

Substituting expression (g) for deflections in the known formulas (101) 
for the bending moments, we obtain 



. mirx 

__ 2qa 2 "ST^ o_ oL m — tanh a m (l + a m tanh a m ) 

ilx — — o J m 3 cos h am am _ tanh a m (a m tanh a m — 1) 



00 

I 



m= 1,3,5,... 

(1 _ „) ««V sinh ™E2 - [2„ + (1 - v)a m tanh «J cosh ^ <*) 
a cl u J 



My 



00 

:qa* VA 



VARIOUS RECTANGULAR PLATES 189 

. rrnrx 

sin 



_ 2qa 2 \ ^ a a m — tanh a m (l + a m tanh g m ) 

m z cosh a m a m — tanh a m (a m tanh a m — 1) 



w = 1,3,5, 



I ( i _ v) tHEM. sinh HHEV + [2 - (1 - „)a m tanh «J cosh ™^1 (i) 
( N a a a ) 

The values of these moments at the center of the plate are 



M x = 



2 9 a 2 V (-1) 



2qa 2 V 

7T 3 L, 



(w-l)/2 



M 



ra 3 cosh a m 

m= 1,3,5, ... 

a m — tanh a m (l + a m tanh a m ) 
a m — tanh a m (a: w tanh a m — 1) 

00 

2ga 2 V (-l)^- 1 )' 2 



[2j> + (1 — v)a m tanh a ro ] 



X 



¥ 7r 3 £j m z cosh a r 

m= 1,3,5, . 

a m — tanh a m (l + a m tanh a m ) 
a m — tanh a m (a m tanh a m — 1) 



[2 — (1 — v)a m tanh a m ] 



These series converge rapidly so that sufficiently accurate values for the 
moments are found by taking only the first two terms in the series. 
Superposing these moments on the moments in a simply supported plate 
(Table 8), the final values of the moments at the center of the plate can 
be represented as follows: 

M x = faqa 2 M y = fcqa 2 (J) 

where /3i and /3 2 are numerical factors the magnitude of which depends on 
the ratio b/a. Several values of these coefficients are given in Table 29. 
Taking the case of a square plate, we find that at the center the 
moments are 

M x = 0.0244ga 2 and M y = 0.0332ga 2 

They are smaller than the moments M x = M y = 0.0479ga 2 at the center 
of the simply supported square plate. But the moments M y at the 
middle of the built-in edges are, as we have seen, larger than the value 
().0479ga 2 . Hence, because of the constraint of the two edges, the magni- 
(ude of the maximum stress in the plate is increased. When the built-in 
sides of a rectangular plate are the longer sides (6 < a), the bending 
moments at the middle of these sides and the deflections at the center of 
the plate rapidly approach the corresponding values for a strip with 
built-in ends as the ratio b/a decreases. 

Plates under Hydrostatic Pressure (Fig. 87). The deflection surface of 
a simply supported rectangular plate submitted to the action of a hydro- 



190 THEORY OF PLATES AND SHELLS 

static pressure, as shown in Fig. 66 (Art. 31), is 



w = 



00 

IT 6 /) Ll 

m= 1 



( _ i)*+i / ^ 2 + a m tanh a m cQsh 



ra° 



V 1 



cosh a m 

+ > 



mTr// 
a 



m ^sinh mxJ/ 



a 



vwy\ 
a J 



cosh a 
The slope of the deflection surface along the edge y = b/2 is 

oo 

/dw\ qoa* V ( 

V 



sin — (A:) 



a 



PV /y=*>/* 






ill 

ra 4 



w+1 



w=l 



[a m - tanh a m (l + a m tanh a m )] sin 



a 



(0 



This slope is eliminated by distributing the moments M y given by series 
(c) along the edges y = ± b/2 and determining the coefficients E m of that 
series so as to make the slope produced by the moments equal and oppo- 
site to that given by expression (0- In this way we obtain 



E. 



2goa 2 ( — l) m+1 am — tanh « m (l + <x m tanh a m ) 
7r 3 ra 3 a m — tanh a m (a m tanh a m — 1) 



Substituting this in series (c), the expression for bending moments along 

the built-in edges is found to be 




Qo 



'////////////// 



V7777777777777 
< o > 

Fig. 87 



T 

I 

I 



{M v )u. 



=±6/2 — 



290a 2 



00 

m== 1 



(-D 



w+1 



Sill 



rrnrx 
a 



m 3 



X>|CVJ 

-4. 



a m — tanh q m (l + a m tanh a m ) , •> 
a m — tanh a m (a m tanh a m — 1) 

The terms in series (m) for Avhich m is even vanish 
at the middle of the built-in sides where x = a/2, 
and the value of the series, as it should, becomes 
equal to one-half that for a uniformly loaded plate 
[see Eq. (/)]. The series converges rapidly, and the value of the bending 
moment at any point of the edge can be readily obtained. Several values 
of this moment together with those of the bending moments along the 
middle line y = of the plate are given in Table 30. 

Concentrated Force Acting on the Plate. 1 In this case again the deflec- 
tion of the plate is obtained by superposing on the deflection of a simply 
supported plate (Art. 34) the deflection produced by moments distributed 

1 See S. Timoshenko, Bavingenieur, 1922, p. 51. 



VARIOUS RECTANGULAR PLATES 



191 



Table 30. Bending Moments in Rectangular Plates with Hydrostatic 
Load, Two Edges Simply Supported and Two Edges Clamped (Fig. 87) 

v = 0.3 





x = a/2, y = 


x = 3a/4, y = 


x = a/2, ?y = fc/2 


x = 3a/4, y - 6/2 


h/a 










M x 


My 


il/x 


My 


Jl/ y 


Af„ 


0.50 


0.007q b 2 


0.021gob* 


0.01 8go&* 


0.029<y fr 2 


-0.042ry b 2 


-0.062^ b 2 


0.75 


O.OlUyob 2 


0.020^ /> 2 


0.018g fc 2 


0.021q b 2 


-0.040g b 2 


-0.045$ 6 2 


1.00 


0.013(/ a 2 


0.017r/ a 2 


0.017ry a 2 


0.015r/«a 2 


-0.035(7 a 2 


-0.035ry a 2 


1.25 


0.021? a 2 


0.02L/ a 2 


0.024^ a 2 


0.019r/ a 2 


-0.045</ a 2 


-0.043g a 2 


1.50 


0.030g a 2 


0.023(/oa 2 


0.031r /0 a 2 


0.020r /0 a 2 


-0 ()51ry a 2 


-0.048g a 2 


2 


0.043</ a 2 


0.024ry a 2 


0.042^ a 2 


0.020r/ a 2 


-0.060ry a 2 


-0.053ry a 2 


OO 


0.063ry a 2 


0.019ry a 2 


0.055ry a 2 


0.017? a 2 


-0.0G3g a 2 


-0.055g a 2 



along the clamped edges. Taking the case of a centrally loaded plate and 
assuming that the edges y = ±6/2 are clamped, we obtain the following 
expression for the deflection under the load: 



w n 



Pb 2 
2ttW 



00 



m = 1,3,5, 



~3 ( tanh a m rf ) 

m 3 \ cosh 2 a m / 

00 

7n V s j_ tanh 2 a m 1 . . 

4 Zy m sinh a m cosh a m + a m J 



w= 1,3,5, . . 



The first sum in the brackets corresponds to the deflection of a simply- 
supported plate [see Eq. (147), page 143], and the second represents the 
deflection due to the action of the moments along the clamped edges. 
For the ratios b/a = 2, 1, £, and •§■ the values of the expression in the 
brackets in Eq. (n) are 0.238, 0.436, 0.448, and 0.449, respectively. 

To obtain the maximum stress under the load we have to superpose on 
the stresses calculated for the simply supported plate the stresses pro- 
duced by the following moments: 



m z = 



00 

-p y a 

Z-/ 4a sinh a 



tanh 



a« 



m= 1,3,5, .. 



-P 



00 



n cosh a m + a m 

[2v + (1 
tanh a m 



v)a m tanh a m ] 



(o) 



m= 1,3,5, , 



4a sinh a m cosh a m + a« 



[2 — (1 — y)a m tanh a m ] 



192 



THEORY OF PLATES AND SHELLS 



Table 31. Correction Bending Moments ati = a/2, y = 0, Due to 
Constraint at y = ±6/2 in Case of a Central Load P (Fig. 71) 

v = 0.3 



b/a 


m x = /3,P 


m v = j8 2 P 


b/a 


m x = PiP 


m v = fid? 




0i 


& 




Pi 


02 





-0.0484 


-0.0742 


1.0 


-0.0505 


-0.0308 


0.5 


-0.0504 


-0.0708 


1.2 


-0.0420 


-0.0166 


G.6 


-0.0524 


-0.0656 


1.4 


-0.0319 


-0.0075 


0.7 


-0.0540 


-0.0580 


1.6 


-0.0227 


-0.0026 


0.8 


-0.0544 


-0.0489 


1.8 


-0.0155 


-0.0002 


0.9 


-0.0532 


-0.0396 


2.0 


-0.0101 


+0.0007 



Putting those correction moments equal to 



(P) 



the numerical factors j8i and j3 2 for various values of the ratio b/a are 
given in Table 31. When the central load P is distributed over the area 
of a small circle or rectangle, we have only to add the moments (p) to 
bending moments obtained for the simply supported plate by means of 
the logarithmical expressions (157) and (167), respectively. The moment 
My at the middle of the clamped edges of a square plate is 

M y = -0.166P 

The calculations show that this moment changes only slightly as the 
length of the clamped edges increases. It becomes equal to — 0.168P 
when b/a = 0.5 and drops to the value of -0.155P when b/a = 1.2.* 
It should be noted that the clamping moment with the numerically 

largest possible value of —P/ir = — 0.3183P is 
produced by a load concentrated near the built-in 
edge of the plate rather than by a central load (see 
Art. 51). In the case of several movable loads the 
influence surface for the clamping moment may be 
used to obtain its maximum value with certainty 
(see Art. 76). 

43. Rectangular Plates with Three Edges Sim- 
ply Supported and One Edge Built In. Let us 
consider a rectangular plate built in along the edge 
y = 6/2 and simply supported along the other edges 
(Fig. 88). The deflection of the plate under any 
lateral load can be obtained by combining the solution for the plate with 

* For further data regarding the plate with two opposite edges built in, see A. 
Pucher, Ingr.-Arch., vol. 14, p. 246, 1943-1944. 



Jl 



Y777777777777. 
< > 



Fig. 88 



T 

.o|CM 



.0|C\J 
l 



VARIOUS RECTANGULAR PLATES 193 

ail sides simply supported, with solution (176) for the case where bending 
moments are distributed along one side of the plate. 

Uniformly Loaded Plates. The slope along the edge y = b/2 produced 
by a uniformly distributed load is 



/dw\ __ 2qa 3 V J_ . rrnrx 
W/v4/2 ~ ir 4 D Li m 4 Sm a 

m = 1,3,5,... 

[a m — tanh a m (l + oc m tanh a m )] ( a ) 

The moments M v = 2E m sin (mirx/a) distributed along the side y = b/2 
produce the slope 1 [see Eq. (176)] 

( ___! ) = __ > — sm E m (cx m tanh 2 a m 

\dy/y=b/2 4ttD Lj m a 

m= 1,3,5, . . . 

— tanh a m + oi m coth 2 a m — coth a m — 2a m ) (6) 

From the condition of constraint these two slopes are equal in magnitude 
and of opposite signs. Hence 

__ Sqa 2 a m — tanh a? m (l + a m tanh a m ) / x 

w "~ 7r 3 m 3 a m tanh 2 a m — tanh a w + oc m coth 2 a m — coth a m — 2a m 

and the expression for the bending moments along the side y = 5/2 is 

00 

(M,W = -^ 2, m« Sm IT 

m= 1,3,5, ... 

a w — tanh a m {\ + <*m tanh a m ) ^ 

2a m — tanh a m (a m tanh a w — 1) — coth a m (a m coth a m — 1) 

Taking a square plate, as an example, the magnitude of the bending 
moment at the middle of the built-in edge from expression (d) is found 
to be 

{My) y ^j2^ai2 = -0.084ga 2 

This moment is numerically larger than the moment — 0.070ga 2 which 
was found in the preceding article for a square plate with two edges built 
in. Several values of the moment at the middle of the built-in side for 
various values of the ratio a/b are given in Table 32. 

Substituting the values (c) of the constants E m into expression (176), 
we obtain the deflection surface produced by the moments of constraint, 
from which the deflection at the center of the plate is 

00 

. a 2 V (-iy m - 1)l2 E m a m tSinha m ,, 

(Wi)*=a/2,v=o - 4^25 L, m> cosh a m W 

m= 1,3,5, ... 
1 Only odd numbers must be taken for m in this symmetrical case. 



194 



THEORY OF PLATES AND SHELLS 



Table 32. Deflections and Bending Moments in a Rectangular Plate 
with One Edge Built In and the Three Others Simply Supported 

(Fig. 88) 
v = 0.3 



b/a 


(u^z.a/a.v-O 


(ilf»)t.«/!,||.«! 


(M x )x-a/i.u-0 


(M v ) Xm . a li,v-0 


00 


0.0130ga 4 /-D 


-0.1259a 2 


0.1259a 2 


0.0379a 2 


2 


0.0093ga 4 /-D 


-0.122ga 2 


0.0949a 2 


0.0479a 2 


1.5 


0.0064ga 4 /£ 


-0.1129a 2 


0.0699O 2 


0.0489a 2 


1.4 


0.0058gaV-D 


-0.109ga 2 


0.0639C 2 


0.0479a 2 


1.3 


0.0050gaV£ 


-0.1049a 2 


0.0569a 2 


0.0459a 2 


1.2 


0.004Zqa*/D 


-0.0989a 2 


0.0499a 2 


0.0449a 2 


1.1 


0.0035ga 4 /Z) 


-0.0929a 2 


0.04l9a 2 


0.0429a 2 


1.0 


0.0028ga 4 /-D 


-0.0849a 2 


0.0349a 2 


0.0399a 2 


1/1.1 


0.0032gb 4 /-D 


-O.O9296 2 


0.03396 2 


0.0439O 2 


1/1.2 


0.0035g&VZ) 


-O.O9896 2 


0.03296 2 


0.04796 2 


1/1.3 


0.0038g6 4 /Z> 


-0.1039fo 2 


0.031 9ft 2 


0.05090 2 


1/1.4 


0. 0040g& */D 


-O.IO896 2 


0.03096 2 


0.05296 2 


1/1.5 


0.0042qb*/D 


-O.IH96 2 


O.O2896 2 


0.05496 2 


0.5 


Q.00i9qb*/D 


-0.1229b 2 


0.02396 2 


O.O6O96 2 





0.0052qb*/D 


-0.1 2596 2 


O.OI996 2 


0.06296 2 



For a square plate the first two terms of this series give 

(wO^a/2,^0 = 0.00127 ^ 

Subtracting this deflection from the deflection of the simply supported 

square plate (Table 8), we find that the deflec- 
tion at the center of a uniformly loaded square 
plate with one edge built in is 




(W) x =a 



:/2,i/=0 



= 0.00279 ^ 



Fig. 89 
of simply supported edges, is (see page 190) 



Values of deflection and bending moments for 
several other values of the ratio a/b obtained in a 
similar way are given in Table 32. 

Plates under Hydrostatic Pressure. If the plate 
is under a hydrostatic pressure, as shown in Fig. 
89, the slope along the edge y = 6/2, in the case 



{dw\ __ qoO* V (-1)"* 1 , 

\dV/*+ii~* A D L m 4 ^ 



— tanh a m — a m tanlr a m ) sin 



rmrx 
a 



if) 



VARIOUS RECTANGULAR PLATES 



195 



The slope produced by bending moments distributed along the edge 
y = 6/2 is 



( -i^i ) = -r^ ) — sin ^^ E m (a m tanh 2 a m — tanh a m 

m= 1 

+ a m coth 2 a m ~ COth a m — 2a m ) (#) 

From the condition of constraint along this edge, we find by equating 
expression (g) to expression (/) with negative sign 

4 9 a 2 (-1)"* 1 



Em = - 



7r 3 m 3 



a m — tanh a m {\ + a m tanh a m ) 



a m tanh 2 a„ 2 — tanh a m + a m coth 2 a m — coth a m — 2a m 
Hence the expression for the bending moment M y along the edge y = b/2 



is 



(M 



00 

, _ 4 9o a 2 V 

m= 1 



(-D 



m+1 



??r 



sin 



rrnrx 



a 



a m — tanh a m {\ + a w tanh a m ) 



2a m — tanh a m (a m tanh a m — 1) — coth a m (a m coth a m — 1) 

This series converges rapidly, and we can readily calculate the value of 
the moment at any point of the built-in edge. Taking, for example, a 
square plate and putting x = a/2, we obtain for the moment at the middle 
of the built-in edge the value 

(M v )y-w2.z-a/2 = -0.042g a 2 
This is equal to one-half the value of the moment in Table 32 for a 
uniformly loaded square plate, as it should be. Values of the moment 
(M y )y =b/2 for several points of the built-in edge and for various values of 
the ratio b/a are given in Table 33. It is seen that as the ratio b/a 
decreases, the value of M y along the built-in edge rapidly approaches the 

Table 33. Values of the Moment M y along the Built-in Edge y = b/2 
of Rectangular Plates under Hydrostatic Load g x/a (Fig. 89) 



b/a 


x = a/4 


x = a/2 


x = f a 


00 


-0.039tf a 2 


-0.062g a 2 


-0.055g a 2 


2 


-0.038g a 2 


-0.061? a 2 


-0.053g a 2 


3 

2 


-0.034? a 2 


-0. 056? a 2 


-0.050g a 2 


1 


-0.0259 a 2 


-0.042g a 2 


-0.040g a 2 


2 
~5 


-0.030g b 2 


-0.056go& 2 


-0.060g b 2 


l 


-O.OSlqob 2 


-O.QGlqob 2 


-0.073g b 2 





-0.031g & 2 


-0.062g & 2 


-0.094?o& 2 



196 



THEORY OF PLATES AND SHELLS 



value — qob 2 x/8a, which is the moment at the built-in end of a strip of 
length b uniformly loaded with a load of intensity q^x/a. 

Now let us consider a plate subjected to a hydrostatic load just as 
before, this time, however, having the edge x = a built in (Fig. 90). 

In applying the method of M. L6vy to this case we take the deflection 

surface of the plate in the form 




B- 



T 
qo 

J. 



YT~r 
v • 

/, -Q|CM 
V I 

2_t 



Fig. 90 



W 



= qtx 
384D 



(16y 4 - 246V + 56 4 ) 



+ 



00 



X m COS 



m7rj/ 



(0 



in which 



m= 1,3,5, ., 



X m = 4 m cosh -£- + J?* 



m7ra; . , rrnrx 
—, — sinh 



+ Cm sinh -y h A 



6 
m7ro: 



cosh 



mirx 



Expression (i) satisfies the differential equation of the bent plate and the 
edge conditions at y = ±6/2 as well. Expanding the expression in 
parentheses in Eq. (t) in the series 



1,5366 



00 



(-1) 



(m-l)/2 



m 6 



cos 



miry 



L mj -^wj 



m = 1,3,5,... 

we obtain the coefficients ^L r 
edges; i.e., 

(iiO-o = I ^ ] - 



from the conditions on both other 



/d 2 w\ 
\dx*) a 



(W) x ^a = 



\dX ) x = a 



U) 



Substitution of the coefficients in expression (i) makes the solution com- 
plete. Deflections and bending moments obtained from the latter equa- 
tion are given in Table 34. 

Table 34. Deflections and Bending Moments in Rectangular Plates 
Clamped at x = a and Carrying Hydrostatic Load (Fig. 90) 

v « 0.3 



b/a 


(w)i_o/2.v-0 


(M x )x-al2.u-0 


(My) x _o/2.»-0 


\JM xjxm. a.v— 


00 


0.0024qoa*/D 


0.029goo 2 


0.009^00* 


-0.067goa 2 


2 


0.0023qoa*/D 


0.029g a 2 


O.Ollgoa 2 


-0.063g a 2 


1.5 


0.0019qoa*/D 


0.026«oa* 


0.013g a 2 


-0.061g a 2 


1.0 


0.0013<? aV£> 


0.019goa* 


O.OI630O 2 


-0.048g a 2 


1 


0.0030qob*/D 


0.028go6* 


0.034q<&* 


-0.071go6 2 


0.5 


0.0045g i»V-D 


0.024g & 1 


0.046go6 2 


-0.084g b* 





0.0065g b 4 /-D 


0.019g„& 2 


0.062go6 2 


-0.125go6 8 



VARIOUS RECTANGULAR PLATES 



197 



CLE 



'A//////////////* 






44. Rectangular Plates with All Edges Built In. 1 In discussing this 
problem, we use the same method as in the cases considered previously. 
We start with the solution of the problem for a simply supported rec- 
tangular plate and superpose on the deflection of such a plate the deflec- 
tion of the plate by moments distributed along 
the edges (see Art. 41). These moments we 
adjust in such a manner as to satisfy the con- 
dition dw/dn = at the boundary of the 
clamped plate. The method can be applied to 
any kind of lateral loading. 

Uniformly Loaded Plates. To simplify our 
discussion we begin with the case of a uniformly 
distributed load. The deflections and the mo- 
ments in this case will be symmetrical with 
respect to the coordinate axes shown in Fig. 91. 
The deflection of a simply supported plate, as 
given by Eq. (139) (page 116), is represented for the new coordinates in 
the following form: 



--->K- 



9 t 

14- 



V JQ|C\J 



2 

y 

Fig. 91 



W = 



4ga 4 



00 

I 



(_l)<m-l)/2 mTX 



m 5 



m= 1,3,5, 



„i7rx ( \ a m tanh a m + 2 , miry 

cos I 1 ~ 1: cos h 

a \ 2 cosh a m a 

1 rrnry . , rrnry\ , . 

+ K r smh ) (a) 

2 cosh a m a a J 



1 For the mathematical literature on this subject see "Encyklopadie der mathe- 
matischen Wissenschaften," vol. 4, art. 25 (Tedone-Timpe), pp. 3 65 and 186. Other 
references on this subject are given in the paper by A. E. H. Love, Proc. London Math. 
Soc, vol. 29, p. 189. The first numerical results for calculating stresses and deflections 
in clamped rectangular plates were obtained by B. M. Koyalovich in his doctor's 
dissertation, St. Petersburg, 1902. Further progress was made by I. G. Boobnov, who 
calculated the tables for deflections and moments in uniformly loaded rectangular 
plates with clamped edges; see his " Theory of Structures of Ships," vol. 2, p. 465, St. 
Petersburg, 1914, and " Collected Papers on the Theory of Plates," p. 144, Moscow, 
1953. The same problem was discussed also by H. Hencky in his dissertation "Der 
♦Spannungszustand in rechteckigen Platten," Munich, 1913. Hencky's method was 
used by I. A. Wojtaszak, /. Appl. Mechanics, vol. 4, p. 173, 1937. The numerical 
results obtained by Wojtaszak in this way for a uniformly loaded plate coincide with 
the values given in Boobnov's table. Further solutions for the same plate and various 
cases of loading are due to H. Leitz, Z. Math. Phys., vol. 64, p. 262, 1917; A. Nadai, 
Z. angew. Math. Mech., vol. 2, p. 14, 1922; A. Weinstein and D. H. Rock, Quart. AppL 
Math., vol. 2, p. 262, 1944; P. Funk and E. Berger, " Federhofer-Girkmann-Fest- 
.s«hrift," p. 199, Vienna, 1950; G. A. Grinberg, Doklady Akad. Nauk. S.S.S.R., vol. 76, 
p. 661, 1951; K. Girkmann and E. Tungl, Osterr. Bauzeitschrift, vol. 8, p. 47, 1953. 
An experimental investigation of the problem is due to B. C. Laws, Phil. Mag., vol. 
24, p. 1072, 1937. Our further discussion makes use of the method developed by 
»S. Timoshenko, Proc. Fifth Intern. Congr. Appl. Mech., Cambridge, Mass., 1938; the 
method is more general than most of those previously mentioned; it can be applied to 
any kind of loading, including the case of a concentrated load. 



198 THEORY OF PLATES AND SHELLS 

where a m = irnrb/2a. The rotation at the edge y = b/2 of the plate is 



00 

/dw\ = 2qa? V 



(_l)(m-l)/2 mTa . 

^-t cos 

m 4 a 



m = 1,3,5, ... 

[a m — tanh a m (l + a m tanh a m )] 

oo 

2ga 3 V (~l) (w " 1)/2 mwx/ a m . , \ , M 

= -^ > ^-r cos ( tanh a m J (6) 

7r 4 D Lj m A a \cosh 2 a m / 

m= 1,3,5, ... 

Let us consider now the deflection of the plate by the moments dis- 
tributed along the edges y = ± b/2. From considerations of symmetry 
we conclude that the moments can be represented by the following series: 

00 

(M v ) y=±bn = Y, (-l) (m - 1)/2 #~cos^p (c) 

m= 1,3,5, ... 

The corresponding deflection Wi is obtained from expression (173) by sub- 
stituting x + a/2 for x and taking ra = 1, 3, 5, .... Then 



00 

a 2 V 7i ( — l) (m_1)/2 mirxfmiry . , rrnr\ 

= - ?TTr; / im 2 u cos I sinh 

27r 2 D Zv ra 2 cosh a m a \ a a 



m= 1,3,5, 



— a m tanh a m cosh j (d) 



The rotation at the edge y = 6/2, corresponding to this deflection, is 

00 

/du>i\ a V v (-l) (m " 1)/2 mirzA , 
I — -1 J = - — = > # m i '- cos — — I tanh a m 

w»= 1,3,5, ... 

+ cosh 2 a m> ) (e) 

In our further discussion we shall need also the rotation at the edges 
parallel to the y axis. Forming the derivative of the expression (d) with 
respect to x and putting x = a/2, we obtain 



(*»l) « V Em V- f^sinh^ 

\ d£ /«-«/2 27rZ> Zv ra cosh a m \ a a 

tn= 1,3,5, . . . 

00 

*— a m tanh a m cosh ) = — -jjr / — rj — 

a ) 4:D JLf cosh 2 a m 

m =1,3,5,... 

(6 sinh a m cosh — — — 2y cosh a m sinh — - ) (/) 
a a / 



VARIOUS RECTANGULAR PLATES 199 

The expression in parentheses is an even function of y which vanishes 
at the edges y = + 6/2. Such a function can be represented by the series 



oo 

I 



Ai cos ^- (g) 

t = l,3,5,... 

in which the coefficients A { are calculated by using the formula 

2 /*+*>/ 2 / 7 . , , miry rt , . , WtttA tVz/ , 

^. = _ / [b smh a m cosh — - — 2y cosh a m smh ) cos -7- ay 

b J -b/2 \ & a / o 

from which it follows that 

At- = o o « t— tt-o cosh 2 a rt 



ra 3 ~ 2 



0/2 ( h l + iiY 

\a 2 + m 2 / 
Substituting this in expressions (gr) and (/), we obtain 

00 00 

m= 1,3,5, ... t = l,3,5,... l a 2 "» m 2 i 

In a similar manner expressions can be obtained for the deflections w 2 
and for the rotation at edges for the case where moments M x are dis- 
tributed along the edges x = ±a/2. Assuming a symmetrical distribu- 
tion and taking 

00 

wi = 1,3,5, . . . 

we find for this case, by using expressions (e) and (h), that 
\dxJ x=a /2 2ttD Z-/ m 6 \ 

m=l,3,5,... 



where jS m = mira/2b, and that 



oo oo 

V«vA-*/« t 2 ^ Z/ m3 Z/ ^fL' + ilY a 

m= 1,3,5, ... t = l,3,5,... \ft2 "r m 2 J 



(*) 



When the moments (c) and (i) act simultaneously, the rotation at the 
<^dges of the plate is obtained by the method of superposition. Taking, 



200 THEORY OF PLATES AND SHELLS 

for example, the edge y = 6/2, we find 



oo 

( dWi dVh\ = a_ V E 

\dy "*" dyJ y=b/2 2tD Li 



(_l)(m-»/2 m7r<T 
cos 



m a 

m= 1,3,5, • , 



( tanh «- + zz£h-J 



^ V £ V wT.™** © 



2 jD6 ^y m 3 ^y /a 2 . t»Y a 

to = 1,3,5,... t- 1,3,5, . 



\6 2 ^ m 2 / 



Having expressions (6) and (Z), we can now derive the equations for 
calculating the constants E m and F m in series (c) and (i) which represent 
the moments acting along the edges of a clamped plate. In the case of 
a clamped plate the edges do not rotate. Hence, for the edges y = + 6/2, 
we obtain 

/3w\ + (dw } + dw 1 \ =Q (m) 

In a similar manner, for the edges x = + a/2, we find 

(M + (*£? + *£A =0 (n) 

\dX/ x= a/2 \dX OX /a-fl/2 

If we substitute expressions (6) and (I) in Eq. (m) and group 1 together 
the terms that contain the same cos (ivx/a) as a factor and then observe 
that Eq. (m) holds for any value of x, we can conclude that the coefficient 
by which cos (iirx/a) is multiplied must be equal to zero for each value 
of i. In this manner we obtain a system that consists of an infinite num- 
ber of linear equations for calculating the coefficients Ei and Fi as follows: 

7r 3 i A \cosh 2 OLi / 

00 

m= 1,3,5,... \^2 + m 2j 

A similar system of equations is obtained also from Eq. (n). The 
constants E h E z , . . . , l^i, F 3 , . . . can be determined in each particu- 
lar case from these two systems of equations by the method of successive 
approximations. 

To illustrate this method let us consider the case of a square plate. 
In such a case the distribution of the bending moments along all sides 
of the square is the same. Hence E { = F { , and the two systems of equa- 

1 It is assumed that the order of summation in expression (I) is interchangeable. 



Ei 
i 



VARIOUS RECTANGULAR PLATES 201 

tions, mentioned above, are identical. The form of the equations is 

00 

TV tanh ^ + ^h^.J + 7 2/ ™ z L 7 ZY 

m= 1,3,5,... y 1 + m 2y 

= 4gg 2 _1 / eg _ , \ 
7r 3 i 4 \cosh 2 oci / 

Substituting the numerical values of the coefficients in these equations 
and considering only the first four coefficients, we obtain the following 
system of four equations with four unknowns Ei 9 Ez, E b , and E 7 : 

1.8033Ei I +0.0764^3 +0.0188# 5 +0.0071# 7 = 0.66772? 



0.0764Ei +0.4045# 3 +0.0330# 5 +0.0159# 7 = 0.012322? ( . 



0.0188^! +0.0330^3 +0.2255i?5 | +0.0163£ 7 = 0.001602? 
0.0071jBi +0.0159^3 +0.0163^5 +0.1558# 7 = 0.000422? 

where 2? = — 4qa 2 /w*. It may be seen that the terms along the diagonal 
have the largest coefficients. Hence we obtain the first approximations 
of the constants E h . . . , E 7 by considering on the left-hand sides of 
Eqs. (p) only the terms to the left of the heavy line. In such a way we 
obtain from the first of the equations E x = 0.37002?. Substituting this 
in the second equation, we obtain E z = —0.03952?. Substituting the 
values of E\ and Ez in the third equation, we find E b = —0.01802?. 
From the last equation we then obtain E 7 = —0.00832?. Substituting 
these first approximations in the terms to the right of the heavy line 
in Eqs. (p), we can calculate the second approximations, which are 
Ei = 0.37222?, Ez = -0.03802?, E b = -0.01782?, E 7 = -0.00852?. 
Repeating the calculations again, we shall obtain the third approxi- 
mation, and so on. 

Substituting the calculated values of the coefficients Ei 9 Ez 9 ... in 
series (c), we obtain the bending moments along the clamped edges of 
the plate. The maximum of the absolute value of these moments is at 
the middle of the sides of the square. With the four equations (p) taken, 
this value is 

\M y \ y = h/2tX= o = \Ei - E* + E b - E 7 \ = 0.0517ga 2 

The comparison of this result with Boobnov's table, calculated with a 
much larger number of equations similar to Eqs. (p), shows that the 
error in the maximum bending moment, by taking only four equations 
(p), is less than 1 per cent. It may be seen that we obtain for the moment 
a series with alternating signs, and the magnitude of the error depends on 
the magnitude of the last of the calculated coefficients Ei, Ez, .... 

Substituting the values of E h Ez, ... in expression (d), we obtain 
the deflection of the plate produced by the moments distributed along 



202 



THEORY OF PLATES AND SHELLS 



Table 35. Deflections and Bending Moments in a Uniformly Loaded 
Rectangular Plate with Built-in Edges (Fig. 91) 

v = 0.3 



b/a 


(w)x=0,i/=0 


(M x ) x =a/2,y=0 


(M 2/ ) x «o,y^/2 


(iW*)x=0.y=0 


(Mj,)j„o,y=0 


1.0 


0.00126qa 4 /D 


-0.0513?a 2 


-0.0513?a 2 


0.0231ga 2 


0.0231ga 2 


1.1 


0.00150qa 4 /D 


-0.0581(/a 2 


-0.0538ga 2 


0.0264ga 2 


0.0231ga 2 


1.2 


0.00l72qa 4 /D 


-0.0639ga 2 


-0.0554(?a 2 


0.0299?a 2 


0.0228ga 2 


1.3 


0.00191qa 4 /D 


-0.0687ga 2 


-0.0563ga 2 


0.0327ga 2 


0.0222ga 2 


1.4 


0.00207qa 4 /D 


-0.0726ga 2 


-0.0568ga 2 


0.0349ga 2 


0.0212ga 2 


1.5 


0.00220qa 4 /D 


-0.0757</a 2 


-0.0570ga 2 


0.0368ga 2 


0.0203ga 2 


1.6 


0.002Z0qa 4 /D 


-0.0780ga 2 


-0.0571ga 2 


0.0381$a 2 


0.0193ga 2 


1.7 


0.002Z8qa 4 /D 


-0.0799ga 2 


-0.0571ga 2 


0.0392ga 2 


0.0182ga 2 


1.8 


0.00245qa 4 /D 


-0.0812?a 2 


-0.0571?a 2 


0.0401ga 2 


0.0174ga 2 


1.9 


0.002A9qa 4 /D 


-0.0822ga 2 


-0.0571ga 2 


0.0407ga 2 


0.0165ga 2 


2.0 


0.00254qa 4 /D 


-0.0829ga 2 


-0.0571ga 2 


0.0412ga 2 


0.0158ga 2 


00 


0.002e>0qa 4 /D 


-0.0833#a 2 


-0.0571ga 2 


0.0417ga 2 


0.0125ga 2 



the edges y = ±b/2. For the center of the plate {x = y = 0) this 
deflection is 



(wi) 



x=y=0 



00 

a 2 V 

2tt 2 /) Li 



E m (-iy m -» 



n 



a„ 



tanh 



m 2 cosh a. 



^= -0.00140^ 



D 



m= 1,3,5, , 







//////////////\ 






Doubling this result, to take into account the action of the moments dis- 
tributed along the sides x = + a/2, and adding 
to the deflection of the simply supported square 
plate (Table 8), we obtain for the deflection at 
the center of a uniformly loaded square plate 
with clamped edges 

(wW = (0.00406 - 0.00280) ^ = 0.00126^ 

(?) 



/ ? 

/ • 

y -q|cm 



7////////////A 



a 
~2~ 



A x>|csj 
V \ 



/. 



Fig. 92 



Similar calculations can be made for any ratio 
of the sides of a rectangular plate. The results 
of these calculations are given in Table 35. l 
Plates under Hydrostatic Pressure. Representing the intensity of the 
pressure distributed according to Fig. 92 in the form 

1 The table was calculated by T. H. Evans; see «/. Appl Mechanics, vol. 0, p. A-7, 
1939. 



VARIOUS RECTANGULAR PLATES 



203 



90= 2 + 2a 

we see that the effect of the term q /2 on the deflections of the plate is 
already given by the previous solution. Thus it remains to consider the 
pressure qox/2a. The deflection surface of a simply supported plate 
carrying such a load is readily obtained by combining the expression (k) 
on page 190 with the expression (a) on page 186. Putting q = — q /2 in 
this latter expression and replacing xbyx + a/2 in both expressions in 
accordance with new coordinates, we obtain the deflection surface 



2qva 4 



y (-i)™ /2+i /. 



2 + a m tanh a„ 



cosh a„ 



cosh 



miry 



a 



m = 2,4,0, 



+ 



1 



cosh a m a 



mrry ^ mry 



wy\ 
a ) 



sin 



rrnrx 



a 



(r) 



symmetrical with respect to the x axis and antisymmetrical with respect 
to the y axis. Consequently, to eliminate the slope along the boundary 
of the plate we have to apply edge moments of the following form: 



(M 



,)«,.= ± ^ (-l)(-»/^.COS^ 



771=1,3,5, , 



(«) 



(M : 



y)>J=±b/a — 7 V 



.l)t»/2-l^ mS i u 



mirX 
a 



7/1 = 2,4,0, ... 



•J///////////fy f 






Load P 









Proceeding just as in the case of the uniformly distributed load, we calcu- 
late the coefficients E m and F m from a system of linear equations. The 
deflections due to the simultaneous action of the 
load qox/2a and the moments (s) must be added, 
finally, to the deflections of the clamped plate 
loaded uniformly with q /2. Numerical re- 
sults obtained by such a procedure are given in 
Table 36. x 

Plates under Central Load. As a third ex- 
ample let us consider the bending of a rectan- 
gular plate with clamped edges under the action 
of a load P concentrated at the center (Fig. 93). 
Again we go back to the case of a simply sup- 
ported plate. Substituting into expression (146) a/2 for £, and x + a/2 

^ee Dana Young, J. Appl. Mechanics, vol. 7, p. A-139, ]940. More extensive 
tables were computed, by means of the method of finite differences, by E. G. Odley, 
J. Appl. Mechanics, vol. 14, p. A-289, 1947. 



L K I 



y 

Fie;. 93 



204 



THEORY OF PLATES AND SHELLS 



Table 36. Deflections and Bending Moments in Rectangular Plates 
with Built-in Edges and Hydrostatic Load (Fig. 92) 

v = 0.3 



b 
a' 


x = 0, y = 


x = a/2, y ■» 


x = -a/2, i/ - 


a; « 0, y - ±6/2 


goa 4 
w~« — 

a 


Mx = Piqoa 2 
Pi 


My = fi2q0CL 2 
02 


Mx = 7igoa 2 
71 


Mx = Tsgoa 2 
72 


Af y = 5goa 2 


0.5 

i 

1.0 

1.5 

00 


0.000080 

0.000217 

0.00063 

0.00110 

0.00130 


0.00198 

0.00451 

0.0115 

0.0184 

0.0208 


0.00515 

0.00817 

0.0115 

0.0102 

0.0063 


-0.0115 
-0.0187 
-0.0334 
-0.0462 
-0.0500 


-0.0028 
-0.0066 
-0.0179 
-0.0295 
-0.0333 


-0.0104 
-0.0168 
-0.0257 
-0.0285 



for x, we arrive at the deflection surface (valid for y > 0) 



w 



Pa 2 V 1 mvx IY + u « m \ 

2^D Li m* C ° S IT LV " m " c-^sh^:; 



m =1,3,5,... 



, m7r?/ . , m7ry 

cosh — - — sinn 

a a 

- ™3L tanh « ro sinh ™* + ^ cosh 



a a 

The angle of rotation along the edge y = b/2 is 



a 



a J 



fdw\ 
V 



<ty / 2/=*>/2 



oo 

= 1,3,5,... 



cos 



m^rrc a* 



tanh 



otn 



a 



cosh 



«« 



(0 



To calculate the bending moments along the clamped edges we proceed 
as in the case of uniform load and obtain the same two systems of Eqs. 
(ra) and (w). The expressions for Wi and w 2 are the same as in the former 
case, and it will be necessary to change only the first term of these equa- 
tions by substituting expression (t) instead of (die/dy)v-*/2 in Eq. (w), 
and also a corresponding expression for (dw/dx) x=a/ 2 in Eq. (n). 

For the particular case of a square plate, limiting ourselves to four 
equations, we find that the left-hand side of the equations will be the 
same as in Eqs. (p). The right-hand sides will be obtained from the 
expression (t), and we find 

0.0764# 3 + 0.0188E 5 + 
" 0.0330S 5 + 



1.8033#i + 
0.0764£i + 
0.0188#i + 
0.007 IE i + 



0.4045^ 3 + 
0.0330^3 + 
0.0159^3 + 



0.2255£ 5 + 
0.0163# 6 + 



0.0071#7 = -0.1828P 
0.0159^7 = +0.00299P 
0.0163£ 7 = -0.000081P 
0.1558^7 = +0.000005P 



VARIOUS RECTANGULAR PLATES 205 

Solving this system of equations by successive approximations, as before, 

we find 

Ei = -0.1025P E z = 0.0263P 
E 5 = 0.0042P E 7 = 0.0015P 

Substituting these values in expression (c), the bending moment for the 
middle of the side y = 6/2 can be obtained. A more accurate calcu- 
lation 1 gives 

(M y )y^,2,*-o = -0.1257P 

Comparing this result with that for the uniformly loaded square plate, 
we conclude that the uniform load produces moments at the middle of 
the sides that are less than half of that which the same load produces if 
concentrated at the center. 

Having the moments along the clamped edges, we can calculate the 
corresponding deflections by using Eq. (d). Superposing deflections pro- 
duced by the moments on the deflections of a simply supported plate, 
we obtain the deflections of the plate with built-in edges. By the same 
method of superposition the other information regarding deflection of 
plates with built-in edges under a central concentrated load can be 
obtained. 2 Thus, if the load P is distributed uniformly over the area 
of a small circle or rectangle, the bending moments at the center of the 
loaded area x = y = can be obtained by combining the results valid 
for simply supported plates [see Eqs. (157) and (167)] with some addi- 
tional moments 

m x = jSiP m y = /3 2 P 

given in Table 37 along with data regarding the maximum deflection of 
the plate and the numerically largest clamping moment. This latter 
moment, however, can reach the value of —P/ir = — 0.3183P, as men- 
tioned on page 192, in the case of a movable load. 

45. Rectangular Plates with One Edge or Two Adjacent Edges Simply 
Supported and the Other Edges Built In. Let us begin with the case of a 
plate simply supported at the edge y = and clamped along the other 
edges (Fig. 94). No matter how the load may be distributed over the 

1 In this calculation seven equations, instead of the four equations taken above, 
were used. 

2 Calculated by Dana Young, /. Appl Mechanics, vol. 6, p. A-114, 1939. To 
obtain the moments with the four correct figures it was necessary to use in this calcula- 
tion seven coefficients E and seven coefficients F in Eqs. (m) and in). Further solu- 
tions of the problem were given by H. Marcus "Die Theorie elastischer Gewebe," 
2d ed. ; p. 155, Berlin, 1932; J. Barta, Z. angew. Math. Mech., vol. 17, p. 184, 1937; 
G. Pickett, J. Appl Mechanics, vol. 6, p. A-168, 1939; C. J. Thorne and J. V. Atanasoff, 
Iowa State Coll. J. Sci., vol. 14, p. 333, 1940. The case was investigated experi- 
mentally by R. G. Sturm and R. L. Moore, J. Appl. Mechanics, vol. 4, p. A-75, 1937. 



206 



THEORY OF PLATES AND SHELLS 



Table 37. Bending Moments at the Middle of Longer Sides and 

Deflections and Additional Moments at the Center of 

Rectangular Plates Loaded at the Center (Fig. 93) 

v = 0.3 









Correction moments 




Pa? 


(M y )x=o,v~bn = yP 










b/a 


D 




(m x ) x =2/«o = /3iP 


(Wl/)*-I/-0 = 02-P 




a. 


7 


Pi 


ft 


1 .0 


0.00500 


-0.1257 


-0.0536 


-0.0536 


1.2 


0.00G47 


-0.1490 


-0.0579 


-0.0526 


1.4 


0.00091 


-0.1G04 


-0.0618 


-0.0517 


1 .0 


0.00712 


-0.1651 


-0.0653 


-0.0510 


1.8 


0.00720 


-0.1667 


-0.0683 


-0.0504 


2.0 


0.00722 


-0.1674 


-0.0710 


-0-0500 


oo 


0.00725 


-0.168 


-0.0742 -0.0484 



given plate sstt, we can consider this plate as one-half of a plate rrlt hav- 
ing all edges clamped and carrying a load antisymmetrical with respect 
to the line ss. The deflections and the bending moments then are zero 
along that line. Thus the problem under consideration is reduced to the 










Fig. 94 

problem already solved in Art. 44. Some numerical data concerning two 
cases of load distribution are given in Table 38. x A more extensive table 

1 The tabulated results are due to Dana Young, J. Appl. Mechanics, vol. 7, p. 
A-139, 1940, and to C. P. Siess and N. M. Newmark, Univ. Illinois Bull., vol. 47, p. 
98, 1950. Y. S. Uflyand used quite a different method in treating this problem; see 
Doklady Akad. Nauk. S.S.S.R., vol. 72, p. 655,H950. 



VARIOUS RECTANGULAR PLATES 



207 



Table 38. Deflections and Bending Moments in Rectangular Plates 
with One Edge Simply Supported and Three Edges Built In 

(Fig. 94) 



Load 



! T nifcrm pressure q . 



Hydrostatic*, pressure r/ ( ,////> 



b/a 





5 





75 


] 





4 




3 




2 







5 





75 


1 






(w)x-,0.!,,,&/2 



0.00440qb 4 /D 
0.00280qb 4 /D 
0.00157qb*/D 
0.002i5qa*/D 
0.002blqa i /D 

0.O02O2(i J)* /D 
O.OOV.Vlq^/D 
0.00014qJ> i /D 



(M,),. 



al2,y=blZ 



-O.Oimqb* 
-0.0730?& 2 
-0.0601 qb* 
-0.0750ga 2 
-0.08379a 2 

-0.0368r/ fc 2 
-0.03447„fo 2 
-0.0287q b 2 



{My) 



y)x = 0,y=b 



-0.1148gfc 2 
-0.0838gfc 2 
-0.0551</6 2 
-0.0571rya 2 
-0.057l5a 2 

-0.0623r/ 6 2 
-0.0484r /0 6 2 
-0.0347g b 2 



of bending moments is given on page 244 in connection with a design 
method for floor slabs. 

The rectangular plate rsut (Fig. 95) with two adjacent edges x = and 
y = simply supported and two other edges clamped can be regarded in 
like manner as an integral part of the plate 
bounded by x = ±a, y = ±b with all edges 
built in. 

Let us consider a load uniformly distributed 
over the area rsut of the given plate. 1 A checker- 
hoard loading distributed over the area 2a by 26 
as shown in Fig. 95 then yields the conditions of a 
simply supported edge along the lines x = and 
// = 0. Thus the problem of bending a plate with 
two adjacent edges simply supported and two 
others clamped is again reduced to the problem, 
already solved in Art. 44, of a plate with all 

edges built in. Calculations show that the numerically largest moment 
is produced near the mid-point of the long side of the plate. The values 
of this clamping moment prove to be — 0.ll80qb 2 for b/a = 0.5 and 
-0.0694g& 2 for b/a = 1.0. The maximum bending moment near the 
center of a square plate has the value of 0.034ga 2 (for v = 0.3) and the 
corresponding deflection is given by 0.0023ga 4 /D. Further numerical 
data regarding bending moments in this case are given on page 243. 




Fig. 95 



1 A modification of Timoshenko's method was applied in handling this case by 
Si ess and Newmark, loc. cit. For use of the energy method see W. B. Stiles, /. Appl. 
Mechanics, vol. 14, p. A-55, 1947. See also M. K. Huang and H. D. Conway, /. Appl. 
Mechanics, vol. 19, p. 451, 1952. 



208 



THEORY OF PLATES AND SHELLS 



46. Rectangular Plates with Two Opposite Edges Simply Supported, 
the Third Edge Free, and the Fourth Edge Built In or Simply Supported. 1 

Let us assume the edges x = and x = a in Fig. 
96 as simply supported, the edge y = b as free, 
and the edge y = as built in. In such a case 
the boundary conditions are 



,-Free 



V///////A 
Fig. 96 



q 



w = 



w = 



d 2 w 
dx 2 



= 



dy 



for # = and # = a 
for ?/ = 



and along the free edge [see Eqs. (112), (113), page 84] 

f y =b L ^ dx 2 dy 



/d 2 W d 2 w\ 







[p + < 2 



») 



1 - 



(a) 
GO 

(c) 



In the particular case of a uniformly distributed load we proceed as in 
Art. 30 and assume that the total deflection consists of two parts, as 
follows : 

where Wi represents the deflection of a uniformly loaded and simply sup- 
ported strip of length a which can be expressed by the series 



W\ = 



4ga 4 

7T 5 D 



oo 

V 1 ■ 

/ —i sin 



rrnrx 



a 



(d) 



m= 1,3,5, 



and Wz is represented by the series 



W2 



00 

I 



Y m sin 



rrnrx 
a 



w 



m =1,3,5,... 



where 



_qa*{ 



Am cosh tzv + Bm am sinh m *y 



a 



a 



a 



+ Cm sinh ™* + D m ^y cosh 



a 



a 



miry\ 
a J 



(/) 



Series (d) and (e) satisfy the boundary conditions (a), and the four con- 
stants in expression (■/) must be determined so as to satisfy the boundary 

1 This case was discussed by Boobnov; see the English translation of his work in 
Trans. Inst. Naval Arch., vol. 44, p. 15, 1902, and his " Theory of Structure of Ships," 
vol. 2, p. 545, St. Petersburg, 1914. It was also discussed by K. Goriupp, Ingr.-Arch., 
vol. 16, p. 77, 1947, and by V. Bogunovic, "On the Bending of a Rectangular Plate 
with One Edge Free," Belgrade, 1953. 



VARIOUS RECTANGULAR PLATES 209 

conditions (6) and (c). Using the conditions (b), we obtain 

Am = ~^ Cw = ""^ (&) 

From the remaining two conditions (c) we find 

» - 4 

m 7r 5 m 5 

(3 + V)(l - ?) cosh 2 |8 m + 2v cosh m 

- Kl - v)$ m sinh j8 m - (1 - v 2 ) 

(3 + v)(l - v) cosh 2 j8 m + (1 - v) 2 /3i + (1 + ^) 2 
4 (fc) 



7r 5 m 5 



(3 + v){\ — j>) sinh /3 m cosh /3 m + v(l + v) sinh j3 m 

- y(l - y )|8 m cosh /3 m - (1 - y) 2 /3 m 

(3 + v){l - v) cosh 2 m + (1 - ^) 2 /3^ + (1 + *0 2 

where j3 m = rrnrb/a. 

Substituting the constants (g) and (/i) in Eq. (/) and using series (e) 
and (d), we obtain the expression for the deflection surface. The maxi- 
mum deflection occurs in this case at the middle of the unsupported edge. 
If the length b is very large in comparison with a, that is, if the free edge 
is far away from the built-in edge, the deflection of the free edge is the 
same as that of a uniformly loaded and simply supported strip of length a 
multiplied by the constant factor (3 — v)(l + v)/(S + v). Owing to the 
presence of this factor, the maximum deflection is larger than that of the 
strip by 6.4 per cent for v = 0.3. This fact can be readily explained if we 
observe that near the free edge the plate has an anticlastic deflection 
surface. 

Taking another extreme case, when a is very large in comparison with 6, 
the maximum deflection of the plate evidently is the same as for a uni- 
formly loaded strip of length b built in at one end and free at the other. 
Several values of the maximum deflection calculated 1 for various values 
of the ratio b/a are given in Table 39. This table also gives the maxi- 
mum values of bending moments which can be readily calculated from 
the expression for the deflection surface. The calculations show that 
(Mx)max occurs at the middle of the unsupported edge. The numerical 
maximum of the moment M y occurs at the middle of the built-in edge. 

The case of the hydrostatic load distributed according to the law 
r/ (l — y/b) can be treated in the same manner as the foregoing case. Let 
the deflection be expressed by 



w 



tt 5 D Z^ m b a Lj " 



a 

1,3/5, .. . w= 1,3,5, .. , 



1 This table was calculated by Boobnov, op. cit. 



210 



THEORY OF PLATES AND SHELLS 



Table 39. Deflections and Bending Moments for a Uniformly Loaded 

Plate with Two Opposite Edges Simply Supported, the Third Edge 

Free, and the Fourth Built In (Fig. 96) 

v = 0.3 



b/a 


Wmax 


x — a/2, y = b 


x = a/2, y = 




M x 


My 




i 

i 

2 

J 

3 
2 

2 

3 

00 


0.1 2596 */D . 
0. 09496 */D 
0.0582qb*/D 
O.OZ35qb*/I) 
0.0ll3qb 4 /I) 
O.OUlqa* /D 
0.0150qa*/D 
0. 01529O */D 
0.0152qa*/D 




0.007890 s 

0.02939O 2 

0.05589O 2 

0.09729O 2 

0.1239a 2 

0.13l9a 2 

0.1339a 2 

0.1339a2 


-O.5OO962 
-O.428962 
-0.31996 2 
-0.22796 2 
-O.II9962 
-0.1249O 2 
-0.1259a 2 
-0.1259a 2 
-0.1259a 2 



in which Y m is of the form (/), only with the constant g instead of q. 

Proceeding as before, we obtain the four constants A m , B m , . . . , D n 

from the boundary conditions (a), (6), and (c). 

If the plate is bent by a load distributed along the free edge, instead oi 

by a load distributed over the sur- 
face, the second of the boundary 
conditions (c) must be modified by 
putting the intensity of the load 
distributed along the free edge in- 
stead of zero on the right-hand side 
of the equation. The particular 
case of a concentrated force applied 
at the free edge of a very long plate 

was investigated (Fig. 97). * It was found that the deflection along the 

free edge can be represented by the formula 




Fig. 97 



(w) 



y=b 



a 



Pb* 
D 



The factor a rapidly diminishes as the distance from the point A of appli- 

1 See C. W. MacGregor, Mech. Eng., vol. 57, p. 225, 1935; D. L. Holl, J. Appl. 
Mechanics, vol. 4, p. 8, 1937; T. J. Jaramillo, J. Appl. Mechanics, vol. 17, p. 67, 1950; 
and K. Girkmann, "Flaehentragwerke," 4th ed., p. 233, Vienna, 1956. The case of a 
cantilever plate having three edges free and carrying a uniformly distributed load was 
discussed by W. A. Nash, J. Appl Mechanics, vol. 19, p. 33, 1952. See also the inves- 
tigation of such a plate by W. T. Koiter and J. B. Alblas with numerical results given 
in Proc. Koninkl. Ned. Akad. Wetenschap. Amsterdam, vol. 60, p. 173, 1957. 



VARIOUS RECTANGULAR PLATES 



211 



ration of the load increases. Several values of this factor are given in 
Table 40. The numerically largest values of the clamping moment pro- 
duced by a load acting at the middle of the free edge of a plate of a finite 
length a are given in Table 41. 1 



Table 40 



X = 





6/4 


6/2 


b 


26 


a = 


0.168 


0.150 


0.121 


0.068 


0.016 



Table 41. Bending Moments M = 0P, at x = 0, y = 0, Due to a Load P 
Acting at x = 0, y = b and the Edges x = ±a/2 Being Simply 

Supported (Fig. 97) 
v = 0.3 



h/a = 


4 


2 


1.5 


1 


2 


0.5 


i 
"5 


0.25 





P = 


-0.000039 


-0.0117 


-0.0455 


-0.163 


-0.366 


-0.436 


-0.498 


-0.507 


-0.509 



The case of a uniformly loaded rectangular plate simply supported 
•.ilong three edges and free along the edge y = b (Fig. 98) can be treated 
in the same manner as the preceding case in 
which the edge y = was built in. It is neces- 
sary only to replace the second of the boundary j 
conditions (6) by the condition i 



[(SMSOL-' 



,'Free 
t 



I Fig. 98 



< knitting the derivations, we give here only the x 

final numerical results obtained for this case. 

The maximum deflection occurs at the middle of 

I ho free edge. At the same point the maximum bending moment M x 

Likes place. These values of deflections w max and (M x ) max are given in 

I lie second and third column of Table 42. 2 The last two columns give 

I he bending moments at the center of the plate. 

Table 43, in a similar manner, contains the values of deflections and 
landing moments produced at the middle of the free edge and at the 
center of the plate by a hydrostatic load. 

47. Rectangular Plates with Three Edges Built In and the Fourth 
Kdge Free. Plates with such boundary conditions are of particular 
interest as an integral part of rectangular tanks or retaining walls. Con- 

1 This table was calculated by V. Bogunovic, loc. ciL See also Art. 78. 

2 This table and Table 43 were calculated by B. G. Galerkin; see Bull. Polytech. 
///.v/., vol. 26, p. 124, St. Petersburg, 1915. 



212 



THEORY OF PLATES AND SHELLS 



Table 42. Deflections and Bending Moments in Uniformly Loaded 

Rectangular Plates with Three Edges Simply Supported and 

the Fourth Edge Free (Fig. 98) 

v - 0.3 



b/a 


* = a/2, J 


) = b 


x = a/2, y = 6/2 


tCmu 


(M*)m.x 


M. 


My 


i 


0.00710gaVZ> 


0.060ga 2 


0.039ga 2 


0.022ga 2 


2 


0.0O96Sqa*/D 


0.083ga 2 


0.055ga 2 


0.030ga 2 


1/1 A 


0.01023qa*/D 


0.088ga 2 


0.059ga 2 


0.032ga 2 


1/1.3 


0.01092qa*/D 


0.094ga 2 


0.064ga 2 


0.034ga 2 


1/1.2 


0.0U58qa*/D 


O.lOOga 2 


0.069ga 2 


0.036ga 2 


1/1.1 


0.01232qa*/D 


0.107ga 2 


0.074ga 2 


0.037ga 2 


1 


0.01286qa*/D 


0.112ga 2 


0.080ga 2 


0.039ga 2 


1.1 


0.01341qa*/D 


0.117ga 2 


0.085ga 2 


0.040ga 2 


1.2 


0.01384ga 4 /D 


0.121ga 2 


0.090ga 2 


0.041ga 2 


1.3 


0.0U17qa*/D 


0.124ga 2 


0.094ga 2 


0.042ga 2 


1.4 


0.01U2qa*/D 


0.126ga 2 


0.098ga 2 


0.042ga 2 


1.5 


0.01462qa*/D 


0.128ga 2 


O.lOlga 2 


0.042ga 2 


2 


0.01507qa*/D 


0.132ga 2 


0.113ga 2 


0.041ga 2 


3 


0.01520qa*/D 


0.133ga 2 


0.122ga 2 


0.039ga 2 


00 


0.01522qa*/D 


0.133ga 2 


0.125ga 2 


0.037ga 2 



Table 43. Deflections and Bending Moments in Hydrostatically Loaded 

Rectangular Plates with Three Edges Simply Supported and the 

Fourth Edge Free (Fig. 99) 

v « 0.3 



b/a 


x = a/2, y = b 


x = a/2, y = 6/2 




w 


M x 


w 


M x 


My 


i 

2 
1 

1.5 
2.0 


0.00230g aV.D 
0.00304goaVD 
0.00368g„a«/D 

0.00347goa*/D 
0.00291g aV.D 



0.0197g„a 2 
0.0265goa 2 
0.0325goa* 
0.0308g a 2 
0.0258goa 2 



0.00135goa 4 /D 
0.00207goaV-D 
0.00313goaV^> 
0.00445goaV-D 
0.00533goa*/D 
0.00651g a 4 /D 


0.0145goa 2 
0.0220g a 2 
0.0331goa 2 
0.0453goa 2 
0.0529goa 2 
0.0625goa 2 


0.0120goa 2 
0.0156goa 2 
0.0214goa 2 
0.0231g a 2 
0.0222goa 2 
0.0187goa 2 



VARIOUS RECTANGULAR PLATES 



213 



sequently, the uniformly distributed and the hydrostatic load must be 
considered first of all in that case. 

Let the boundary of the plate be clamped at y = and x = ±a/2 
:uid free along y = b (Fig. 100). Assuming first a uniformly distributed 
load of intensity q, the expression for deflections may be taken in the form 



w = W\ + Wz + Wz 



The expressions for 



Wi 



md 



00 

ir 5 Z> Z/ 

W= 1 

oo 



(_l)<m-l)/2 mirX 
^-r- — COS 



ra c 



m= 1,3,5, . .. 



F m (-l) (m - 1)/2 COS 



a 



m7ra; 
a 



(a) 



(6) 



(c) 



m = 1,3,5, . . . 



contained in Eq. (a) are identical with expressions (d) and (e) of the 
preceding article if one considers the new position of the origin. 



x-4 



,'Free 



< a > 

Fig. 99 




! 



K 2 



,'Free 



! ^ Z 

x 7777777777777/. *% 



\ 



2 ** 
Fig. 100 



A suitable form for the additional deflections w 3 due to the additional 
constraint on the edges x — ±a/2 is 1 



II) 



00 

,5, . . . 

5? £ ( 



W~ Fn -W smh -W) sin "a 



A . 

- 1 Sll 



n= 1,3,5, 

qa q 



Gnsmh^ + H^cosh™^ 



a 



a 



m= 1,3,5, . . . 



+ I m ™* sinh 



rrnry\ 
~a~) 



cos — ~~ (a) 



a a / a 

m which F n , . . . , I m are some constants and y n = mra/4b. 

1 This method of solution essentially is due to Goriupp, op. cit., p. 153 1948. 
See also W. J. Van der Eb, Ingenieur, vol. 26, p. 31, 1950. 



214 



THEORY OF PLATES AND SHELLS 



As w 3 = for y = and x = ± a/2, the boundary conditions still t 
be satisfied by deflections (d) are the following: 






Vh , d 2 W 3 \ 
dy 2 V dz 2 ) y =b 

\dy 







A =0 

A=0 



^ +(2 - F )-^L 



Jl/=6 



at/ 3 ' x " '' da; 2 dy] y=b 
|" d(tt>i + w 2 + «?») "[ _ Q 

L ^X Jx=±o/2 







<« 



! 



,- Free 



i 






2 H 




Now we expand all noneireular functions of x contained in expression (a 
in a series of the form Sa m cos (rrnrx/a) and all similar functions of y ii 
a series of the form 26 n sin (rnry/2b). A set of linear equations for F n 
G ny . . . , Im is then readily obtained from conditions (e). Solving tin 
equations we are able to express those unknown constants by the knowi 

values of A m , . . . , D m (see page 209). 

In the case of a hydrostatic pressure 
acting in accordance with Fig. 101, we hav< 
to superpose solution (i) of the preceding 
article on the solution of form (d) and 
besides that, to proceed as indicated above 
Whatever the load, the problem can alsc 
be handled 1 by the method of finite differ- 
ences (see Art. 83). Numerical values ol 
Tables 44 and 45 are computed essentially 
by that procedure. 1 
48. Rectangular Plates with Two Opposite Edges Simply Supported 
and the Other Two Edges Free or Supported 
Elastically. Let us consider the case where the 
edges x = and x = a (Fig. 102) are simply sup- 
ported and the other two edges are supported by 
elastic beams. Assuming that the load is uni- 
formly distributed and that the beams are iden- 
tical, the deflection surface of the plate will be 
symmetrical with respect to the x axis, and we 
have to consider only the conditions along the 
side y = 6/2. Assuming that the beams resist 
bending in vertical planes only and do not resist 
torsion, the boundary conditions along the edge y = 6/2, by using Eq 
(114), are 



Fig. 101 



^il 



/Elost. beam or free 




Fig. 102 



( 



d 2 W 

dy 2 



+ V 



d 2 w \ 

dx 2 ) 







W=*/2 



d [S?+p-'>b5,L.-( 



EI—\ 



(a) 



1 See A. Smotrov, "Solution for Plates Loaded According to the Law of Trapeze/ 1 
Moscow, 1936. 



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216 



VARIOUS RECTANGULAR PLATES 217 

where EI denotes the flexural rigidity of the supporting beams. Pro- 
ceeding as in Art. 46, we take the deflection surface in the form 

W = Wi + V)2 (6) 

00 

. 4ga 4 Y* 1 . rrnrx , N 

where W\ = — r^ / — r sin (c) 

7r b D jL/ m h a ' 

m = 1,3,5, . . . 

00 

and w 2 = y Y m sin (d) 

ra = l,3,5, . .. 

From symmetry it can be concluded that in expression (/) of Art. 46 we 
must put C m = An = and take 

Y m = q -£ ( Am cosh ^ + 5 m rHEk sinh ^ ( e ) 

The remaining two constants A m and B m are found from the boundary 
conditions (a), from which, using the notations 

rmrb EI x 

we obtain 

4*> 
/l m (l — i/) cosh a m + 5 m [2 cosh a m + (1 — v)a m sinh a m ] 



mV 5 



— -4 m [(l — j>) sinh a m + mirX cosh a m ] + B m [(l +* v) sinh a\ n 

4X 
— (1 — v)a m cosh a m — rmr\dm sinh a m ] = — t— ; 

771 V* 

Solving these equations, we find 
4 



-4 m = 



m 6 7r 5 



(/) 



j/(1 + v) sinh a TO — *>(1 — v)a m cosh a m — ra7rX(2 cosh a m 

+ a™ sinh a m ) 

(3 + v)(l — *>) sinh a m cosh a m — (1 — i^) 2 a m + 2?n7rX cosh 2 a m 

_ 4 v(l — v) sinh a m + rmr\ cosh a m . v 

m " ^V (3 + j0(1 - iO sinh a m cosh a m - (1 - ^) 2 a m ^ 

+ 2ra7rX cosh 2 a m 

The deflection surface of the plate is found by substituting these values 
of the constants in the expression 



00 



w = w\ + uh = ^r ) I / c + A m cosh 



a 

m= 1,3,5, 

, n miry . , wnry\ . m7rx /tN 

+ B m — - smh — ~ ) sin (h) 

a a J a 



218 



THEOKY OF PLATES AND SHELLS 



If the supporting beams are absolutely rigid, X = oo in expressions (/ 
and (g), and A m and B m assume the same value as in Art. 30 for a plat* 
all four sides of which are supported on rigid supports. 

Substituting X = in expressions (/) and (g), we obtain the values o: 
the constants in series (h) for the case where two sides of the plate are 
simply supported and the other two are free. 

Except for the case of very small values of X the maximum deflectior 
and the maximum bending moments are at the center of the plate. 
Several values of these quantities calculated for a square plate and foi 
various values of X are given in Table 46. * 

Table 46. Deflections and Bending Moments at the Center of a 

Uniformly Loaded Square Plate with Two Edges Simply 

Supported and the Other Two Supported by Elastic 

Beams (Fig. 102) 

v = 0.3 



X = EI/aD 


^max 


CM*)m« 


l^yjmax 


00 


0.00406ga 4 /Z> 


0.0479ga 2 


0.0479ga 2 


100 


0.00409ga 4 /Z) 


0.0481ga 2 


0.0477ga 2 


30 


0.00416ga 4 /Z> 


0.0486ga 2 


0.0473ga 2 


10 


0.00434ga 4 /Z) 


0.05000a 2 


0.0465ga 2 


6 


0.00454ga 4 /Z> 


0.0514ga 2 


0.0455ga 2 


4 


0.00472ga 4 /Z> 


0.0528ga 2 


0.0447ga 2 


2 


0.00529ga 4 /Z) 


0.0571ga 2 


0.0419ga 2 


1 


0.00624ga 4 /2> 


0.0643ga 2 


0.0376ga 2 


0.5 


0.00756ga 4 /Z> 


0.0744ga 2 


0.0315ga 2 





0.01309ga 4 /Z> 


0.1225ga 2 


0.0271ga 2 



The particular case X = of a plate with two opposite edges simply supported and 
the other two free deserves some consideration. As Table 47 2 shows, the deflections 
and the largest moments of such a plate loaded uniformly differ but little from the 
deflections and moments of a plate bent to a cylindrical surface. 

49. Rectangular Plates Having Four Edges Supported Elastically or Resting on 
Corner Points with All Edges Free. Let us consider a plate subjected to a uniform 
pressure and supported along the boundary by four flexible beams. All beams are 
supposed to have rigid supports at the corners of the plate, and two beams parallel to 
each other may have the same flexural rigidity (Fig. 103). 

1 The table was calculated by K. A. Calisev, Mem. Inst. Engrs. Ways Commun., 
St. Petersburg, 1914. More recently the problem was discussed by E. Mliller, 
Ingr.-Arch., vol. 2, p. 606, 1932. The tables for nonsymmetrical cases are calculated 
in this paper. Various cases of rectangular and continuous plates supported by 
flexible beams were discussed by V. P. Jensen, Univ. Illinois Bull., 81, 1938. 

2 These results are due to D. L. Holl, Iowa State Coll. Eng. Exp. Sta. Bull. 129, 1936. 
For the case of a concentrated load see also R. Ohlig, Ingr.-Arch., vol. 16, p. 51, 1947. 
Both authors also discuss the effect of clamping the supported edges. 



VARIOUS RECTANGULAR PLATES 



219 



Table 47. Deflections and Bending Moments in Uniformly Loaded 

Rectangular Plates with the Edges x = 0, x = a Simply 

Supported and the Other Two Free (Fig. 102) 

v - 0.3 





X 


= a/2, y = 





x = a/2, y = ±6/2 


h/a 


qa 4 

Oil 


M x = friqa* 
/3, 


My = faa* 
6 


qa* 

«2 


M x = Piqa 2 


0.5 
1.0 
2.0 

00 


0.01377 
0.01309 
0.01289 
0.01302 


0.1235 
0.1225 
. 1235 
0.1250 


0.0102 
0.0271 
0.0364 
0.0375 


0.01443 
0.01509 
0.01521 
0.01522 


0.1259 
0.1318 
0.1329 
0.1330 



By writing the deflections in the form 



384D( 7 + 5) 



[ 7 (16x 4 - 24a 2 * 2 + 5a 4 ) + 5(Wy 4 - 246V + 5fc 4 )] 

mrx rnry 
cosh — - cos — - 
b b 



Xniry mrx V^ 
A n cosh cos + > #„ 
a a l^i 

"O . n7r?/ n?ra: V^ . 

+ / C„7/ sinh — - cos + > DnX smh 

/ j/ a a 21/ 



ri7rx nwy . N 
— - cos — (a) 



r 



^^■-=^ 



o. 



K 2 



X 
i 

i 

i 

Xl|CM 



"t 



-O|0s) 



where 6/7 and A n , . . . , A t are some constants and n = 1, 3, 5, . . . , we satisfy the 
differential equation A Aw = q/D of the plate and also the conditions of symmetry. 1 
Next, let us develop the algebraic and the hyperbolic 
functions contained in expression (a) in cosine series. 
Then, using for x = a/2 and y = ft/2 the edge condi- 
tions similar to conditions (a) of the preceding article, 
we arrive at a set of equations for the constants 
.■!„, . . . , D n of expression (a). 

Making, in particular, 6/7 = and EJ h = «>, we 
would arrive at the solution of the problem already 
discussed in Art. 48. 

Let us consider now the bending of a square plate 
(a = b) supported by four identical beams. We have 
l hen, by symmetry, 6/7 = 1, and A n = B n and 
( ' n = D n . The unknown coefficients A n are eliminated 
l»y equating to zero the edge moments. Taking, then, 

only four terms (n = 1, 3, 5, and 7) in series (a), we arrive at four linear equations for 
<' h C h C 5 , and C 7 . The results of numerical calculations carried out in this way are 
iliven in Table 48. 

1 This method of solution is due to B. G. Galerkin; see his " Collected Papers ; ,, vol. 
L\ p. 15, Moscow, 1953. The boundary conditions under consideration are easily 
realizable and thus appropriate for the verification of the theory by tests. 
N. Dimitrov, Bauingenieur, vol. 32, p. 359, 1957. 



~a-4 



y 

Fig. 



103 



See 



220 



THEORY OF PLATES AND SHELLS 



Table 48. Deflections and Bending Moments of a Square Plate with 

Fcur Sides Supported Elastically (Fig. 103) 

v - 0.25 





X = 


= 0, y = 


x « 0, y - 6/2 


_ EI 








7 ~ aD 


qa A 

iv = a 

D 


M x = M v - Piqa* 


M x = ptqa* 




a 


(81 


fi* 


00 


0,00406 


0.0460 





100 


0.00412 


0.0462 




50 


0.00418 


0.0463 




25 


0.00429 


0.0467 


0.0002 


10 


0.00464 


0.0477 


0.0024 


5 


0.00519 


0.0494 


0.0065 


4 


0.00546 


0.0502 


0.0085 


3 


0.00588 


0.0515 


0.0117 


2 


0.00668 


0.0539 


0.0177 


1 


0.00873 


0.0601 


0.0332 


0.5 


0.01174 


0.0691 


0.0559 





0.0257 


0.1109 


0.1527 



In the particular case of EI = we have a square plate carrying a uniformly dis- 
tributed load and supported only at the corners. The value of v has but little influence 
on the deflections and moments at the center of the plate; its effect on the edge 
moments is more considerable. Taking, for example, v = 0.3 the values given in the 
last line of Table 48 for v = 0.25 should be replaced by 0.0249, 0.1090, and 0.1404 
respectively. 1 

The problem of bending of a centrally loaded square plate fixed only at the corners 
has also been discussed. 2 If the load P is distributed uniformly over a small area of a 
rectangular or circular outline, an expression can be deducted 3 for moments taking 
place at the center of the loaded area. Taking, for example, a square loaded area 
u by u, those moments for v = 0.3 can be expressed in the form 



M x = M t 



.(.. 



1034 log - + 
u 



.129 V 



(&) 



Having this solution and also the solution for the uniformly loaded square plate 
supported at the corners, the problem shown in Fig. 104a can be treated by the method 
of superposition. It is seen that if a square plate with free edges is supported by the 

1 See H. Marcus, "Die Theorie elasticher Gewebe, ,, 2d ed., p. 173, Berlin, 1932; 
various cases of plates fixed at points were discussed by A. Nddai, Z. angew. Math. 
Mech., vol. 2, p. 1, 1922, and also by C. J. Thorne, J. Appl. Mechanics, vol. 15, p. 73, 
1948. 

* See Marcus, ibid. 

8 See S. Woinowsky-Krieger, Ingr.-Arch., vol. 23, p. 349, 1955. 



VARIOUS RECTANGULAR PLATES 



221 



uniformly distributed reactions, the bending moments at the center are obtained by 



subtracting from expression (6) the value M x = M y 
uniformly loaded square plate supported at the 
corners and having v = 0.3. In this way we obtain 



= 0.1090ga 2 , given above for the 



M: 



= My = ( 



1034 log - + 0.020 
u 



) 



P (c) 



valid for v = 0.3. The distribution of bending 
moments along the middle line of the footing slab 
is shown in Fig. 104b for u/a =0.1 and u/a = 0.2. 
A uniform distribution of the pressure may be as- 
sumed for a very rigid footing slab resting on soft 
subgrade. More general hypotheses regarding the 
law of distribution of that pressure will be postu- 
lated in Chap. 8. 

50. Semi-infinite Rectangular Plates under 
Uniform Pressure, The deflection surface and the 
stress distribution near the short side of long 
rectangular plates are practically the same as those 
at the ends of semi-infinite plates, as shown in Fig. 
105. It is mainly for this reason that the simple 
theory of these latter plates deserves consideration. 




MMMMMMMM r P /q2 



Fig. 104a 
Let the load be uniformly dis- 




forg-=0.1 



Fig. 1046 

tributed over the area of the entire plate and let the edges x = 0, x = a be simply 
supported. 1 

1 The following solutions of the problem are due to A. N&dai; see his book "Elas- 
tische Platten, ,; p. 72, Berlin, 1925. 



222 THEORY OF PLATES AND SHELLS 

The deflection surface of the plate may be expressed by 



in which 



W = W\ + w 2 



(a) 



W] 






(5) 



m= 1,3,5, 



is the particular solution of the equation AAw = g/D, 5 being the intensity of the load, 
and 



Wo 



? 4qa 4 V* / wirA _ i . rrnrx 



(c) 



m = 1,3,5, .. 



is a solution of the equation AAw = 0, yielding zero deflections at y = 00. The 

coefficients A m and 2? m , which are still at our 
disposal, must be determined so as to satisfy 
the respective conditions along the edge y = 
of the plate. The following three cases may 
be considered. 

The edge y = is simply supported (Fig. 105a) . 
The particular conditions to be fulfilled are 
w = and dhv/dy 2 = f or y « 0. Substitu- 
tion of the series expressing w = w\ + w 2 in 
those conditions yields the values A m = — 1/m 6 
and B m = A m /2 for the coefficients. Thus we 
arrive at the deflection surface 



D 





'////// 







X 




X 




< Q — -5» 




<— a— * 




<-— a— > 



(a) 



(b) 
Fig. 105 



(c) 



t» = M)i — 



4qa* 
7*D 



i o+^) 



1 + -^— J : — sin 



ra° 



(d) 



?n= 1,3,5, .. 



in which w\ is given by Eq. (fr). 

Of particular interest are the bending moments M y of the plate. Along the middle 
line x = a/2 of the plate we have, by differentiation, 



My = 



vqa 2 
~8~ 



+ 



4qa 2 



t b-*?-] 



g—micyla 



m* 



(_l)(m-l)/2 



(e) 



m= 1,3,5, ..t 



Making use of the condition dMJdy = and taking into account the first term of the 
rapidly convergent series, we conclude that M y becomes a maximum at 



a 1 + v 

y = -- 

7T 1 — V 



Table 49 gives the largest values of bending moments together with the largest values 
of the edge reactions V v and the forces R acting downward at the corners of the plate. 
It should be noted that the value 0.0364ga 2 exceeds by 45 per cent the value 
0.0250ga 2 of the largest moment M y of an infinitely long plate, the value of Poisson's 
constant being the same in both cases. 



VARIOUS RECTANGULAR PLATES 



223 



Table 49. Largest Bending Moments and Reactions of a Uniformly 
Loaded Semi-infinite Plate with All Edges Simply Supported 

(Fig. 105a) 



V 


(M x )m&x 


(iwy)iuax 


( r y) max 


# 


0.2 
0.3 


0.1250ga2, x = -> y = « 

a 
0.1250ga 2 , x = -> y = « 


0.0364ga2. s = -> 2/ = 0.48a 

a 
0.0445ga 2 , x = -j i/ = 0.59a 


0.520oa, x = -j 2/ = 
0.502ga, a: = -* 2/ = 


0.1085oa2 
0.0949ga2 



The edge y = is fr?/iM m (Fig. 1056). Following the general procedure described 
:i.bove, but using this time the edge conditions w — 0, dw/dy = on y =0, we obtain, 
instead of expression (d), the result 



w = W\ 



00 

4ga 4 V 114- m?r?/ 1 



e - miry la ^ TUttX 

sin 



771° 



a 



(/) 



m= 1,3,5, . . 

in which Wi again is given by Eq. (b). The corresponding bending moment 



M v 



vqa 



2 



4ga 2 



8 + . 



t I""" 5 ?- 1 -'] 



e -mrryla m7rX 

sin 

ra 3 a 



w 



m= 1,3,5, ... 



becomes a maximum at x = a/2 and ?/ = 2a/7r(l — *>)• Assuming v = 0.3 we obtain 
// = 0.91a and (M v )max = 0.0427aa 2 , whereas the assumption of v = 0.2 yields the 
values of 0.0387ga 2 and y = 0.80a, respectively. It can be shown, also, that the 
variation of the clamping moments along the short side y = of the plate obeys the 
simple law 

(M y ) y=0 = - | (ax - x 2 ) 

Observing that at large values of y the deflection surface of the plate can be assumed 
cylindrical, we have there 

M x = | (ax - x 2 ) My = v\ (ax - x 2 ) 

'Thus, the distribution of the edge moments (g) is identical with the distribution of the 
moments M x across the plate at y = oo but with opposite sign. 

The edge y = is free (Fig. 105c). If the conditions prescribed at y = are 



d 2 w dhv 

v 1 =0 

dx 2 dy 2 



d z W , d 3 W 

h (2 - v) — = 

dy 3 dx 2 dy 



then, making use of expressions (a), (6), and (c), we arrive at the deflection surface 

4pqa 4 



w = W\ + 



(3 + v)** 



Zy \1 - *> a / 



e -mir v /a ^ mwX 

— — : — sin — (A) 



ra° 



a 



224 



THEORY OF PLATES AND SHELLS 



The deflection and the bending moment M x are largest at the middle of the free edge. 
It can be proved that 

3 - v 

(w)y-0 = ~ 77^—- : 7 Wi 



and 



(1 - „)(3 + p) 

nur \ (3 - y )(l + v) 

(M x ) ym . = — (Af»)i 

6 + v 



w\ and (ilfaOi being the deflections and the moments of an infinite simply supported 

plate. We have therefore 



J 






* 


/ 






/ 


1 








/ 






/ 


', 


*-s— 


<-%--> 


'/ 


'/ 


1 


2 


t> 


/ 






/ 


/ 






y 


/ 






y 


A 






\ 



(Afx)max = 



(3 -*)(!+ v) qa* 
3 + v 8 



w = 



y 

Fig. 



4ga 4 



106 



As a last example, leading to a different form of solution, 
let us consider a uniformly loaded semi-infinite plate with 
the edge y = simply supported and the edges x = ±a/2 
built in (Fig. 106). The solution can be obtained by sub- 
stituting 6 = oo in a suitably chosen expression for the 
deflections of a finite rectangular plate simply supported on 
the edges y = 0, b and clamped on the edges x = ±a/2. 
The result of such a derivation, which is omitted here, is 



tD 



I sinh - + - cosh - ) cosh si 

\ 2 2 2/ a a 



sinh - sinh — 
2 a 



sinh p + fi 



sin- 



Pv 



dp 



(0 



Differentiating expression (i) and observing that 



sin — djS 

a 7T 

^ = 2 



for y > 



we obtain 




sin — dp 

a q 



P 



D 



Thus the differential equation for bending of plates is satisfied. It can be shown that 
the required boundary conditions at y = and x = ±a/2 are also satisfied by solu- 
tion (i). 

The expressions for the bending moments of the plate again involve infinite integrals, 
which can be evaluated. Once more the moments M y are of interest. Assuming, for 
example, v = 0.2, we arrive at a value of (M y ) max = 0.0174ga 2 , occurring at y « 0.3a, 
whereas the moment M y = vqa 2 /24t of an infinite plate does not exceed 0.00833(ja 2 for 
the same value of v. 

It should be noted that the properties of the semi-infinite plates can be used as 
a basis for calculating the deflections and bending moments of finite rectangular 
plates with simply supported or built-in edges in any given combination. 1 

1 For this approach to the theory of rectangular plates see W. Koepcke, Ingr.-Arch., 
vol. 18, p. 106, 1950. 



VARIOUS RECTANGULAR PLATES 



225 



61. Semi-infinite Rectangular Plates under Concentrated Loads. Assuming the 
edges x = and x = a of the plate to be simply supported, let us consider, regarding 
the third side (y = 0), the following two cases: (1) the edge y = is simply supported, 
and (2) the edge y = is clamped. 

The edge y = is simply supported (Fig. 107). Assuming that the given load P is 
applied at point x = £ , y = 77 (Fig. 107), we first consider an infinite plate supported 
only at the edges x = and x = a. In order to use the method of images (see page 
156), we assume a second load —P acting at the point x = £ , y = —77 of the infinite 
plate. The line 7/ = becomes then a nodal line of the deflection surface of the plate. 
Thus the required bending of the semi-infinite plate is obtained by superposing the 



«S Q > 




Load-P 

1 
1 




1 

1 
Load+pl 

? 







[ 



Fig. 107 

deflections [see Eq. (148), page 145] produced in the infinite plate by both concen- 
trated loads. In this way we arrive at the deflection surface 



W\ = 



Pa 2 
2tt*D 



7 e m-K , I . niTi 

/ 1 + — (77 - y) sin — 

771= 1 



m7r£ . 77171-2 

sin 



Pa* 



7 6 m7r . 7?i7r^ . m7ra; 

/ j H (v + y) \ sin sin 

Jm—j m z |_ a j a a 

w= 1 



or, after some rearrangement, 



u>i 



00 

Pa 2 V e~ mT1 > la \ ( min\ . , rrnry rrnry rmryl . 

= — r / H I sinh cosh si 

?r 3 D L-1 7?i 3 L\ a / a a a J 

771= 1 



mvrf . 77171-2? 

sin sin 



(a) 
The 



un expression valid for < y < 77 and yielding w\ = 0, d 2 w\/dy 2 = at 7/ =0. 
deflections in the range of y > 77 may be obtained in a similar manner. 

If we distribute the single load over a small area, the moments M x at the center of 
that area and the corresponding deflections prove to be smaller than those of an 
infinite plate without the transverse edge at y = 0. But the moment M y is again an 



226 



THEORY OF PLATES AND SHELLS 



exception. Let us write this moment in the form M y = M y0 + m y , where M y0 is the 
moment of the infinite plate. The correction m y , representing the effect of the load — P 
in Fig. 107, is then readily found by means of the second of the equations (151) (see 
page 145). Assuming, for example, v = 0.3 we obtain m y = 0.0065P as the largest 
value of the correction, the corresponding position of the load being given by x = a/2, 
y = 0.453a. 







'//////////// 



c _ a > 



P 
o- 



i 

i 
i 

i 
-1 



—e-> 



Fig. 108 

The edge y = is built in (Fig. 108). We begin with the calculation of the slope 
of the elastic surface (a) at y = 0, for which differentiation gives 



00 

(dwA _ Pr, V 
\ dy /jm, *D jLf 



e -mirr,/a m ^ mwX 

sin sin 

m a a 



(b) 



Next let us submit the simply supported semi-infinite plate to couples distributed 
along the edge y = in accordance with the law 



(J* ,)_-/(*) 



00 

■I 

m = l 



E m sin 



mwx 



The corresponding deflections, vanishing at y = oo , we take in the form 



2 = /j ( Am + B my)e- mry,a sin ^^ 



(c) 



m— 1 



The coefficients A m and B m in this expression are readily obtained from the conditions 
This yields A m = 0, B m = E m a/2m7rD, and, finally, 



V)2 



00 

2,rZ> // 

m= 1 



E m e" mTV,a . mTrx 

sm 

m a 



<<o 



VARIOUS RECTANGULAR PLATES 227 

Since we have to eliminate the slope (6), the edge condition is 

(dtl7i\ /dw 2 \ 

— ) +(-7-1 =0 (/) 

Substitution of expressions (6) and (e) in Eq. (/) gives 

2Prj t . VlTrt 
Em = e -mirr,/a sm 

a a 

and expression (e) becomes accordingly 



W ye . nix, 
_i_ / sin 

vu A f m a 



Pyn \ e a . nix^ . mirx 
W2 — — / sin sin (^) 

TTj) 

The deflection surface of the semi-infinite plate clamped on y = then is given by 

w — W\ + w^ (h) 

where w\ denotes expression (a). As for the series (g), it can be represented in a 
closed form. We have only to express the sine functions contained in (g) in terms of 
the exponential functions 

g±(m.7r£r7a) and gdb(wir»t/a) 

and to observe the expansion 

log a ± #) = ±# - — ± y - • • • 

If we proceed in this manner, expression (g) finally appears in the simpler form 

cosh - (y + 77) - cos - (x - £) 

47Tl>/ 7T 7T 

COSh - (i/ + 77) — COS -(#+(;) 

a a 

The value of the clamping moments at y = is readily obtained by differentiation of 
expression (z), and the result is 

(Afy)y-O = - 7T Sinh ~ I ~ " """" ^ W 

la a I 7TT? 7T 7r77 7T 

cosh cos -(# — £) cosh — — cos -(# + £) 

a a a a 

When the concentrated load approaches the built-in edge y = 0, the value given by 
expression (j) tends to zero in general. If, however, £ = x and 77 — > simultaneously, 
then Eq. (j) yields 

1 2irX \ 

P 7T S T" \ P 

(M y ) ymmQ = - lim ( -~- coth ^~ ^— — J = (k) 

2a 2a tttj 2wx I ir 

cosh — — cos / 

a ° / ,-0 

If, finally, 77 = 0, the moment M y becomes zero. 



228 



THEORY OF PLATES AND SHELLS 



In conclusion let us consider a single load P (Fig. 109) uniformly distributed over a 
straight-line segment of some length u. The moment caused by such a load at the 
mid-point of the built-in edge is readily found by means of expression (j). Substitute 



°ff 



'////////// 



i 
i 

i 

p* 

i 

i 

i 



i 



•SL/ 
2 

a 



.j 



2. 



Fig. 109 

ing x = a/2 and P d£/u for P in this expression and integrating we obtain for the 
required moment 



en* \ 2PlJ - x. ^ 
\M y ) Xm .a/2,v-o = sinn — 

au a 



2Prj 



arctan 




sinh — 



sin — d{ 
a 



,2xi7 2tt{ 

cosh 1- cos 

a a 



Table 50 gives the position of the load producing the numerically largest clamping 
moment and the value of that moment for various values of the ratio u/a. 

Table 50. Largest Clamping Moments atx = a/2 Due to a Single Load 
Distributed over a Length u (Fig. 109) 



u/a 





0.1 


0.2 


0.4 


0.6 


0.8 


1.0 


v/a 
MJP 



-0.318 


0.147 
-0.296 


0.203 
-0.275 


0.272 
-0.237 


0.312 
-0.204 


0.321 
-0.172 


0.343 
-0.143 



CHAPTER 7 

CONTINUOUS RECTANGULAR PLATES 



62. Simply Supported Continuous Plates. Floor slabs used in build- 
ings, besides being supported by exterior walls, often have intermediate 
supports in the form of beams and partitions or in the form of columns. 
In the first case we have to deal with proper continuous plates; in the 
case of columns without intermediate beams we have to deal with flat 
slabs. The floor slab is usually subdivided by its supports into several 




k ~2 _>k "2 ^2^2 ^2 ^2 *" 
(a) 



f F^ 1 



pmn^ 



(b) 



(C) 

Fig. 110 



panels. Only continuous plates with panels of rectangular shape will be 
considered in this chapter. 

We begin with a case allowing a rigorous solution by methods already 
used in the foregoing chapter. A rectangular plate of width b and length 
ai + a 2 + a 3 , supported along the edges and also along the intermediate 
lines 55 and tt, as shown in Fig. 110, forms a simply supported continuous 
plate over three spans. We suppose that the intermediate supports 
neither yield to the pressure in the transverse direction nor offer any 
resistance to the rotation of the plate with respect to the axes ss and tt. 
With these assumptions, the bending of each span of the plate can be 
readily investigated by combining the known solutions for laterally 
loaded, simply supported rectangular plates with those for rectangular 
plates bent by moments distributed along the edges. 

229 



230 THEORY OF PLATES AND SHELLS 

Let us begin with the symmetrical case in which 

CL\ = d2 = 03 = a 

and the middle span is uniformly loaded while the side spans are without 
load (Fig. 1106). Considering the middle span as a simply supported 
rectangular plate and using expression (b) of Art. 44 (see page 198), we 
conclude that the slope of the deflection surface along the edge x 2 = a/2 is 

dX2/x 2 =a/2 

-% t '-=^ -Tub:— '«•) 



( 



(a) 
m~ u \uusii" p m / 

w = 1,3,5, ... 

where j8 m = rmra/2b. Owing to the continuity of the plate, bending 
moments M x are distributed along the edges x 2 = ±a/2. From sym- 
metry it is seen that these moments can be represented by the following 
series: 

oo 

(M x ) x ^ ±an = ^ (- 1)<— »/«#. cos ^ (b) 

m= 1,3,5, . .. 

The deflections W\ produced by these moments can be obtained from 
Eq. (173), and the corresponding slope along the edge x* = a/2 [see 
Eq. (e), page 198] is 



(p) _ » v. « 



(-1) 



(w-l)/2 



m =1,3,5,, 



miry 
cos —j-^ 
6 



( tenh *« + ^) <«> 



From the condition of continuity we conclude that the sum of expres- 
sions (a) and (c) representing the slope of the plate along the line x* = a/2 
must be equal to the slope along the same line of the deflection surface 
of the plate in the adjacent span. Considering this latter span as a 
simply supported rectangular plate bent by the moments (6) distributed 
along the edge Xz = —a/2, we find the corresponding deflection w* of 
the plate by using Eq. (176) (see page 185), from which follows 



[cosh/? m V 



m= 1,3,5, ... 



mry (-1) <*-»/« 

its ; _ _ l 



W2== 4^D L ^ C ° S 6 



/J m tanh /3 m cosh — j— j— smh — j— J 

■ gar*; (^ coth ^ sinh -6 r cosh nr J J w 



CONTINUOUS RECTANGULAR PLATES 231 

The corresponding slope along the edge x* = — a/2 is 

00 

Xdzzjzt—ait 4ttD Li m 

m= 1,3,5, ... 

cosh ^ (tanh fc + coth fc + J^-^) (e) 
The equation for calculating the coefficients E m is 



(dw\ (dwi\ __ /dw 2 \ 

dX2/x 2 =a/2 ! \dX2/x 2 =a/2 \dX Z /*,=_ «/2 



Since this equation holds for any value of y, we obtain for each value of 
m the following equation: 

2?6 3 1 / jS m + U * \ h E m(, ^ Q , Pm \ 

^D m* V^PX ^ tanh ^ J " 2^ ^ (, ^ + ^^ J 

= A; — ("tanh /3 m + coth ft. + — §=- -1^-^ (/) 

4:ttD m \ cosh 2 /? m sinh 2 fi m J 

from which 

8gb 2 p m — tanh j8 m cosh 2 fi m 

7r 3 ?n 3 3 tanh /3 m cosh 2 j8 m + coth fi m cosh 2 /3 m + 3/3 m — /3 m coth 2 /3 m 

(g) 



E 



It is seen that E m decreases rapidly as m increases and approaches the 
value — 2g6 2 /7r 3 m 3 . Having the coefficients E m calculated from (g), we 
obtain the values of the bending moments M x along the line tt from 
expression (6). The value of this moment at y = 0, that is, at the 
middle of the width of the plate, is 

00 

(ilf*)x 2 =±a/2,^0 = ^ iU-l) (m ~ 1>/2 

m= 1,3,5, . . . 

Taking, as an example, b = a, we have @ m = rmr/2, and the formula (#) 
gives 

^ = - *2* 0.1555 tf* = - ^ 0.0092 E b = - ^ 0.0020 

7T 3 7T 3 7T 

(M x ).,-±«/2.j^o = -0.0381ga 2 

The bending moments at the center of the middle span can be readily 
obtained by combining bending moments of a simply supported plate, 
bent by uniform load, with moments corresponding to the deflections W\. 
Taking, for example, a = b and v = 0.2, which is a convenient value for 



232 THEORY OF PLATES AND SHELLS 

concrete, we get for the first of these moments the values of 
(Af,) = (M v )o = 0.0479 X ^| ga 2 = 0.0442 9 a 2 

(see Table 8, page 120) and for the second moments the values 
(M,)i = -0.0067ga 2 and {M y ) l = -0.0125ga 2 

Therefore 

(M x ) X2 =o, y =o = 0.0375ga 2 (M^o.^o = 0.0317ga 2 

If a side span is uniformly loaded, as shown in Fig. 110c, the deflection 
surface is no longer symmetrical with respect to the vertical axis of sym- 
metry of the plate, and the bending moment distributions along the lines 
55 and tt are not identical. Let 



00 

(M.)«„ - ^ (~ lY m - 1) "E m cos ^ 

m= 1,3,5, ... 

00 

(M.)«/i= Yf (-l) (m - 1)/2 *Vcos^ 

m= s 1,3,5, ... 



(h) 



To calculate the coefficients E m and F m we derive two systems of equa- 
tions from the conditions of continuity of the deflection surface of the 
plate along the lines ss and tt. Considering the loaded span and using 
expressions (a) and (e), we find that the slope of the deflection surface 
at the points of the support ss, for ai = a* = a 3 = a, is 

(£) _2g V (-l)'r""cos^(-^-tan M.) 

\dxi/ Xl=a /2 ir*D Li m* b \cosh 2 /3 m Kw / 

1,3.5, . . . 

V g . <- 1 ^-'" co S h^( t anhfe 

1,3,5,... 

Considering now the middle span as a rectangular plate bent by the 
moments M x distributed along the lines ss and tt and given by the series 
(/&), we find, by using Eq. (175) (see page 184), 

wi = 1,3,5, . • • 

(jsfs + tenh "") + ( *~ - '■> ( coth *■ - sfes;)l w 



wi — 1,3,5, , 

b 
4ttD 

m= 1,3,5, . 



CONTINUOUS RECTANGULAR PLATES 233 

From expressions (i) and ( j) we obtain the following system of equations 
for calculating coefficients E m and F m : 

A m ^ft + E m (B m + C m ) = -B m (E m + F m ) - C m (E m - F m ) (fc) 

where the following notations are used: 

The slope of the deflection surface of the middle span at the supporting 
line tt, by using expression (j), is 

(*•) _ » V '- 1 )'-'" cos^U. + F.) 

m = 1,3,5,... 

(s&E + tanh "") + (F " " £J ( coth fc " SEFfc)] 

This slope must be equal to the slope in the adjacent unloaded span 
which is obtained from expression (c) by substituting F m for E m . In this 
way we find the second system of equations which, using notations (I), 
can be written in the following form: 

B m (E m + F m ) + C m {F m - E m ) = - (B m + C m )F w (m) 

From this equation we obtain 

Fm = Em 2{B m + C m ) (n) 

Substituting in Eq. (fc), we find 

l? - A 8ga 2 2(J? m + On) , . 

iW ~ ^ w 7r 3 m 3 (C m - B m y - 4(B m + C m ) 2 w 

Substituting in each particular case for A m , J5 m , and C m their numerical 
values, obtained from Eqs. (Z), we find the coefficients E m and F m ; and 
then, from expressions (h), we obtain the bending moments along the 
lines ss and tt. Take, as an example, b = a. Then |8 m = m7r/2, and we 
find from Eqs. (I) 

Ai = -0.6677 Bi = -1.1667 Ci = -0.7936 
A z = -0.9983 Bz = -1.0013 C 3 = -0.9987 

For m larger than 3 we can take with sufficient accuracy 



234 THEORY OF PLATES AND SHELLS 

Substituting these values in Eq. (o), we obtain 



8qa 2 



E x = - 



0.1720 



Es = 



Sqa 2 

7T 3 3 3 



0.2496 



E 6 = - ^ 0.2500 

7T 3 5 3 



= -0.0424?a 2 



The moment at the middle of the support ss is 
(M x ) Xl=a/2fV=0 = Ei - Ez + E s - • • 
l'or the middle of the support tt we obtain 

(M x ) Xt=a/2 , v ^ = Fx - F z + F b - • • • = 0.0042ga 2 

Having the bending moments along the lines of support, the deflections 
of the plate in each span can readily be obtained by superposing on 
the deflections produced by the lateral load the deflections due to the 
moments at the supports. 

The bending moments in the panels of the continuous plate can be 
obtained in a similar manner. Calculating, for example, the moments 
at the center of the middle span and taking v = 0.2, we arrive at the 

values 



<U -• °i+i H 



i T' 



y 



Ts 

i 

I 



i+l 



*t+i 






(M x ) x ^o,y=o = -0.0039ga 2 
(M v ) Xt=0 ,y= = -0.0051ga 2 



-A. 



The equations obtained for three 
spans can readily be generalized and 
expanded for the case of any number of 
spans. In this way an equation similar 
to the three-moment equation of con- 
tinuous beams will be obtained. 1 Let 
us consider two adjacent spans i and 
i + 1 of the length a,- and Of + i, respec- 
tively (Fig. 111). The corresponding values of the functions (/) are 
denoted by A^, B l m , C l m and A% 1 , B^ 1 , CJ+ 1 . The bending moments along 
the three consecutive lines of support can be represented by the series 



U- — ai— >U- -a l+r -H 
JL__i a) 

Fig. Ill 



Mi" 1 



Mi 



M i + 1 



oo 

l 

1,3,5 

00 

■ 2 

m= 1,3,5, . 

00 

- I 

:n~ 1,3,5,. 



(-l)^/ 2 ^ 1 ^^ 



m= 1,3,5, . 



(-l)<™-i>/*i^cos 



miry 



(-iy™-i)/2 E i+i cos 



miry 



1 This problem in a somewhat different way was discussed by B. G. Galerkin: see his 
"Collected Papers," vol. 2, p. 410, Moscow, 1953. 



CONTINUOUS RECTANGULAR PLATES 235 

Considering the span i + 1 and using expressions (a) and (j), we find 
\dxu-i/,.„- (.,„)/! » 4 -D Zv m 4 6 

ra = 1,3,5, . . . 
ra = 1,3,5, . . . 

- (i*^ 1 + vjcit 1 ] ( P ) 

In the same manner, considering the span i, we obtain 



00 

dw\ _ 2 gi b* V (-1) ( '- 1)/2 pni5 rmry , t 



m— 1,3,5 



6 V (-1)^-^ m^ 
+ 1^ Li m co& ~5~ [Cj& - +iiJ ^ 

m = 1,3,5, . . . 

+ (EU - Eir^CU (g) 
From the condition of continuity we conclude that 



/ dw \ _ /dttA 

\3.l' i+ i/ x . +1 =_(o. fl )/2 \3.Ti/x,=a,/2 



Substituting expressions (p) and (g) in this equation and observing that 
it must be satisfied for any value of y, we obtain the following equation 
for calculating E*- 1 , E % m , and Eg 1 : 

EtKBL - Ci) + EKBi + Ci + B^ 1 + C^) 

+ #i +1 (^ +1 - Cl +1 ) = - JJ£i (ffi+iA^ + gMD (177) 

Equations (fc) and (?n), which we obtained previously, are particular cases 
of this equation. We can write as many Eqs. (177) as there are inter- 
mediate supports, and there is no difficulty in calculating the moments 
at the intermediate supports if the ends of the plate are simply supported. 
The left-hand side of Eq. (177) holds not only for uniform load but also 
for any type of loading that is symmetrical in each span with respect to 
the x and y axes. The right-hand side of Eq. (177), however, has a 
different value for each type of loading, as in the three-moment equa- 
tion for beams. 

The problem of continuous plates carrying single loads can be treated 
in a similar manner. In the particular case of an infinite number of 
equal spans with a single load applied at any point of only one span, the 
deflection of the plate may be obtained by resolving an equation with 



236 



THEORY OF PLATES AND SHELLS 



EL as functions of the 



finite differences for the unknown coefficient 
index i. 1 

If the intermediate supports are elastic, the magnitude of the coeffi- 
cients EU is governed by the five-term equations, similar to the five- 
moment equations of the theory of continuous beams. 2 The torsional 
rigidity of supporting beams, tending to reduce the rotations of the plate 
along the support, can also be taken into account in considering the 
bending of continuous plates. 3 

As the simplest example of a continuous plate carrying a concentrated load, let us 
consider an infinitely long plate simply supported along the sides x = 0, x = a, con- 
tinuous over the support y = 0, and submitted to a concentrated load P at some point 
x = J, y = t? (Fig. 112a). The load and boundary conditions under consideration 
can be readily satisfied by superposition of cases shown in Fig. 1126 and c. In the 

case of Fig. 1126 each panel of the plate is 
simply supported along the line y = 0, and 
the elastic surface is given by the expression 
±wi/2, in which the sign must be chosen 
according to whether y is greater or less 
than zero, Wi denotes the deflections (a) of 
Art. 51, and \y\ < \rj\. In the case shown in 
Fig. 112c, each panel is clamped along the 
edge y = 0, and the corresponding deflec- 
tions are w/2 9 w being given by expression 
(h) in Art. 51. We have therefore 




w = W\ + 



V)2 



f or rj > y > 



Fig. 112 



W2 

= — for y < 



and the moments along the edge y = become equal to one-half of the clamping 
moments of a semi-infinite plate with one edge built in, these latter moments being 
given by expression (j) of Art. 51. 

53. Approximate Design of Continuous Plates with Equal Spans. 4 
The layout of a floor slab usually involves continuity not only in one 
direction, as assumed in Art. 52, but rather in two perpendicular direc- 
tions. A continuous slab of this kind is shown in Fig. 113. The spans 
and the thickness of the plate are equal for all rectangular panels. Each 

1 See S. Woinowsky-Krieger, Ingr.-Arch., vol. 9, p. 396, 1938. 

* Continuous plates on elastic beams were considered by V. P. Jensen, Univ. Illinois 
Bull. 81, 1938, and by N. M. Newmark, Univ. Illinois Bull. 84, 1938. 

8 See K. Girkmann, "Flachentragwerke," 4th ed., p. 274, Vienna, 1956. 

4 The method given below is substantially due to H. Marcus; see his book "Die 
vereinfachte Berechnung biegsamer Flatten," Berlin, 1929. The coefficients of Tables 
51 to 56 are, however, based on solutions considered in Chap. 6 and on the value of 
Poisson's ratio v = 0.2, whereas Marcus uses for the same purpose a simplified theory 
of rectangular plates and assumes v = 0. 



CONTINUOUS RECTANGULAR PLATES 



237 



panel may carry a dead load go and, 
possibly, a live load p, both distri- 
buted uniformly over the area of 
the panel, the largest intensity of 
the load being q = q + p. 

Let us begin with the computation 
of bending moments at the inter- 
mediate supports of the floor plate. 
Calculations show that these mo- 
ments depend principally on the 
loading of the two adjacent panels, 
and the effect of loading panels 
farther on is negligible. It is justi- 
fiable, therefore, to calculate the 
moments on supports by assuming 
the load q uniformly distributed 
over the entire floor slab (Fig. 114a). Neglecting, at first, the rotations 
of the plate along the intermediate supports, each panel in Fig. 114a will 
have the same conditions as a rectangular plate clamped along the inter- 
mediate supports and simply supported at the external boundary of the 
floor slab. 



t 


© 

r 

y 1 


< 

© 

' r 

u y 1 


© | 


© 

r 


© 

I — x ' 
y' 


© 

r 

y 1 




w 
z 


© 

r 

© 

r 

y 1 


w 

2 





Fig. 113 





1 


w 


m 


if 




M 
M 


r -Qo + P 




1 








1 





(o) 








I 


j 
i 




1 








— 


+ 


P 
2 




1 





(c) 



Fig. 114 



238 THEORY OF PLATES AND SHELLS 

The maximum bending moments for plates with such boundary con- 
ditions have been tabulated (see Tables 51 to 56). Six possible combi- 
nations of simply supported and built-in edges of a rectangular plate are 
shown at the head of these tables. The direction of the x and y axes in 
each panel of the slab (Fig. 113) must be chosen in accordance with Figs. 
116 to 121; span a must be measured in the direction of the x axis and 
span b in the direction of the y axis of the respective panel. The six 
cases shown in Figs. 116 to 121 may be numbered 1 to 6, and the corre- 
sponding indices are attached to the coefficients of Tables 51 to 56. 

To illustrate the application of the tables, let us calculate the bending 
moment at the middle of the support tw (Fig. 113). We calculate for 
this purpose the clamping moment of both panels adjacent to the sup- 
port. For panel 2 we have to use the formula 

Mt M = hql* (a) 

and Table 52, I being the smaller of spans a and b of the panel. In a 
similar manner we obtain the clamping moment of panel 6 from the 
expression 

Me* = y*ql 2 (P) 

by making use of Table 56. The moment in question now is given with 
sufficient accuracy by 

M tw = £(M 2 „ + M 6x ) (c) 

and the moments on other intermediate supports are obtainable in a sim- 
ilar manner. 

It should be noted that Eq. (c) expresses nothing else than a moment- 
distribution procedure in its simplest form, i.e., a procedure in which the 
"carried-over" moments from other supports, as well as any difference 
in the stiffness values of both adjacent panels, are neglected. Such a 
simplified procedure is far more justified in the case of a continuous plate 
than in the case of a continuous beam. 

Next, let us consider the bending moments at the center of panel 6 
(Fig. 113) as an example. The load distribution most unfavorable for 
these moments can be obtained by superposition of loads shown in Fig. 
1146 and c. 

The contribution of the uniformly distributed load go + p/2 to the 
values of the moments is obtained by use of Table 56, which gives 

ML = a 6 (q + f ) Z 2 M', v = 0. (go + |) Z 2 (<*) 

I denoting the smaller of both spans of panel 6. 

Let us consider now the effect of the checkerboard loading as shown in 
Fig. 114c. The boundary conditions of each panel here are the same as 



CONTINUOUS RECTANGULAR PLATES 



239 



those of a simply supported plate, and the. moments at the center are 
readily computed by means of Table 51 for case 1. The load +p/2 
acting in panel 6 yields 



va-fr 



mz = Q£p 



6y 



(e) 



and the largest moments at the center of panel 6 are 



M< x 



+ M' t ' m 

ML + M" 



M' 6x 



6y 



(/) 



62/ 



In order to calculate the largest nega- 
tive moments at the same point we 
have only to alter the sign of the load 
in Fig. 1 14c. Still using results (d) and 
(e), we then have 



M 6 , = M\ x - M& 

M'e' y 



Mey = M' 



(g) 



6?/ 



As a second example of the application of 
the approximate method, let us compute 
the bending moments of the continuous 
plate shown in Fig. 115, which was treated 
rigorously in Art. 52. 

First we choose the direction of the x and 
;// axes in accordance with Figs. 117 and 118. 
Assuming next a load q = q + p uniformly 
distributed over the entire surface of the 
plate (Fig. 1156) and using the coefficients 
given in Tables 52 and 53 for cases 2 and 3, 
with b/a = 1, we obtain at the center of the 
support ss the moment 



M ss = - 



0.0840 + 0.0697 



(o) 



© 



© 



0->L° JloJ.0 J< olo. 



© 



O|0sJ 
i 

o|c\j 



w liuunnj(iiiijnx 



(c) 



7V 



3L 



*""A -T q ° 



(d ) p-q uHnjtrn^ qo 



TP . . ; I 



le) UUIUIlUllllj [ 



Mp 



f^o 



(f) 



4 W W 



f 



* * + + 



Up 



=H-1p 



Fig. 115 



(go + p)a 2 = -0.0769(g + p)a 2 



(h) 



the procedure being the same as in the foregoing example [Eq. (c)]. Using the rigorous 
solution, the numerically largest moment at ss is produced by the load distribution 
shown in Fig. 115c. Superposing the bending moment obtained on page 231 upon 
those calculated on. page 234, the exact minimum value of the moment M ss proves 
to be 



or 



M ss = -[0.0381 (g + v) + 0.0424(g + p) 
M ss = - (0.0805g + 0.0763p)a 2 



0.0042g ]a 2 



(0 



Putting, for instance, g = g/3, p = 2g/3, the result (i) yields -0.0777ga 2 as compared 
with the value -0.0769ga 2 obtained by the approximate method. 

Finally, let us calculate the largest bending moment at the center of the middle 
panel, the most unfavorable load distribution being such as shown in Fig. 115d. 



240 



THEORY OF PLATES AND SHELLS 



c — *J 



<*,£ 



Fig. 116 

Table 51. Bending Moments for Uniformly Loaded Plates in Case 1 
v = 0.2, I = the smaller of spans a and 6 





Center of plate 




b/a 


M x = aiql* 


My - |8,<tf2 


Factor 




«i 


01 




n 

KJ 


0.0250* 


0.1250 




0.5 


0.0367 


0.0999 




0.6 


0.0406 


0.0868 




0.7 


0.0436 


0.0742 


qb* 


0.8 


0.0446 


0.0627 




0.9 


0.0449 


0.0526 




1.0 


0.0442 


0.0442 
0.0449 




1.1 


0.0517 




1.2 


0.0592 


0.0449 




1.3 


0.0660 


0.0444 




1.4 


0.0723 


0.0439 




1.5 


0.0784 


0.0426 




1.6 


0.0836 


0.0414 


qa* 


1.7 


0.0885 


0.0402 




1.8 


0.0927 


0.0391 




1.9 


0.0966 


0.0378 




2.0 


0.0999 


0.0367 




cc 


0.1250 


0.0250f 





M 



|M 



= 0.0364g6 2 at 0.486 from the short edge. 
max = 0.0364ga 2 at 0.48a from the short edge. 



Combining the load in accordance with Fig. 115e and / and using the coefficients a 
and of Tables 53 and 51, we arrive at the following expressions for these moments: 



M x = 0.0216 Uo + |J + 0.0442^ o* = (0.0216<? + 0.0329p)c ! 
M y = 0.0316 (««+|)+ 00442 2 K = (° 0316 9o + 0.0379p>' 



V) 



CONTINUOUS RECTANGULAR PLATES 



241 



: 



*! 



'////////////, 



s 

a,/3 



Fig. 117 

Table 52. Bending Moments for Uniformly Loaded Plates in Case 2 
v = 0.2; I — the smaller of spans a and b 









Middle of 






Center of plate 


fixed edge 




b/a 








Factor 


M x = ai ql 2 


M v = /3 2 «Z 2 


My = 5 2 ql* 






«2 


Pi 


5 2 







0.0125 


0.0625 


-0.1250 




0.5 


0.0177 


0.0595 


-0.1210 




0.6 


0.0214 


0.0562 


-0.1156 




0.7 


0.0249 


0.0514 


-0.1086 


qb* 


0.8 


0.0272 


0.0465 


-0.1009 




0.9 


0.0294 


0.0415 


-0.0922 




1.0 


0.0307 


0.0367 


-0.0840 






1.1 


0.0378 


0.0391 


-0.0916 




1.2 


0.0451 


0.0404 


-0.0983 




1.3 


0.0525 


0.0415 


-0.1040 




1.4 


0.0594 


0.0418 


-0.1084 




1.5 


0.0661 


0.0418 


-0.1121 


qa* 


1.6 


0.0722 


0.0414 


-0.1148 




1.7 


0.0780 


0.0408 


-0.1172 




1.8 


0.0831 


0.0399 


-0.1189 




1.9 


0.0879 


0.0390 


-0.1204 




2.0 


0.0921 


0.0382 


-0.1216 




00 


0.1250 


0.0250* 


-0.1250 





* iWma* = 0.0387ga 2 at 0.80a from the built-in edge. 

It is of interest to verify the foregoing approximate values by use of the results 
obtained on pages 232 and 234. Distributing the load again as shown in Fig. 115d 
and interchanging the indices x and y in the results mentioned above, we have 

M x = 0.0317(^0 + V)a 2 - (0.0051 + 0.0051)? a 2 

= (0.0215g + 0.0317p)a 2 (k) 

M v = O.O375(0<> + p)a 2 - (0.0039 + 0.0039)g a 2 
= (0.0297g + 0.0375p)a 2 

Setting again q = g/3 and p = 2g/3, we obtain for the moments the exact values 
of 0.0283ga 2 and 0.0349r/a 2 , respectively. Eqs. (j) yield for the same moments the 
approximate values of 0.029 Iqa 2 and 0.0358ga 2 . 



242 



THEORY OF PLATES AND SHELLS 



r— °— i 



8 
oc-,0 



v;///>v ;///;. 



Fig. 118 



Table 53. Bending Moments for Uniformly Loaded Plates in Case 3 
v = 0.2, I = the smaller of spans a and b 





Center of plate 


Middle of 
fixed edge 




b/a 






Factor 


M x = a 3 ql* 


M y = /3 3 gi* 


M v = 8 3 ql* 




a 3 


03 


&* 







0.0083* 


0.0417 


-0.0833 




0.5 


0.0100 


0.0418 


-0.0842 




0.6 


0.0121 


0.0410 


-0.0834 




0.7 


0.0152 


0.0393 


-0.0814 


qb 2 


0.8 


0.0173 


0.0371 


-0.0783 




0.9 


0.0196 


0.0344 


-0.0743 




1.0 
1.1 


0.0216 
0.0276 


0.0316 
0.0349 


-0.0697 
-0.0787 






1.2 


0.0344 


0.0372 


-0.0868 




1.3 


0.0414 


0.0391 


-0.0938 




1.4 


0.0482 


0.0405 


-0.0998 




1.5 


0.0554 


0.0411 


-0.1049 


qa 2 


1.6 


0.0620 


0.0413 


-0.1090 




1.7 


0.0683 


0.0412 


-0.1122 




1.8 


0.0741 


0.0408 


-0.1152 




1.9 


0.0795 


0.0401 


-0.1174 




2.0 


0.0846 


0.0394 


-0.1191 




00 


0.1250 


0.0250f 


-0.1250 





* Afmax = 0.0174g& 2 at 0.306 from the supported edge, 
t M m&x = 0.0387ga 2 at 0.80a from the built-in edge. 

The largest error of the approximate method ensues from the fact that the largest 
positive moments do not always occur at the center of the panel. This is especially 
far from being true in the case of distinctly oblong rectangular panels. If 6, for 
example, is much larger than a, the largest moment M v occurs near the short side of 
the rectangular plate. Some values of these largest moments are given in footnotes to 
the tables, and they should be considered as the least possible values of the corre- 
sponding columns, regardless of the actual ratio b/a. 

It should be noted, finally, that in the unsymmetrical case 4 neither M , nor M y 



CONTINUOUS RECTANGULAR PLATES 



243 






1 y / 

b ^r 
1 ^ 

1 y y 
1 ^ 
4 V. 



8 






y 0.4 a 
Fig. 119 

Table 54. Bending Moments for Uniformly Loaded Plates in Case 4* 
v = 0.2, I = the smaller of spans a and b 





Center of plate 


Middle of fixed edge 


Atx = 0.1a 
y = 0.16 




b/a 












Factor 


M z = cnql* 


M v = /3 4 g/ 2 


M x = 7<^ 2 


My = «4^ 2 


iWma x = etql* 






en 


04 


74 


«4 


«4 




0.5 


0.0191 


0.0574 


-0.0787 


-0.1180 


0.0662 




0.6 


0.0228 


. 0522 


-0.0781 


-0.1093 


0.0570 




0.7 


0.0257 


0.0460 


-0.0767 


-0.0991 


0.0501 


qb* 


0.8 


0.0275 


0.0396 


-0.0746 


-0.0882 


0.0430 




0.9 


0.0282 


0.0336 
0.0281 


-0.0715 
-0.0678 


-0.0775 
-0.0678 


0.0363 
0.0305 




1.0 


0.0281 




1.1 


0.0330 


0.0283 


-0.0766 


-0.0709 


0.0358 




1.2 


0.0376 


0.0279 


-0.0845 


-0.0736 


0.0407 




1.3 


0.0416 


0.0270 


-0.0915 


-0.0754 


0.0452 




1.4 


0.0451 


0.0260 


-0.0975 


-0.0765 


0.0491 




1.5 


0.0481 


0.0248 


-0.1028 


-0.0772 


0.0524 


qa? 


1.6 


0.0507 


0.0236 


-0.1068 


-0.0778 


0.0553 




1.7 


0.0529 


0.0224 


-0.1104 


-0.0782 


0.0586 




1.8 


0.0546 


0.0213 


-0.1134 


-0.0785 


0.0608 




1.9 


0.0561 


0.0202 


-0.1159 


-0.0786 


0.0636 




2.0 


0.0574 


0.0191 


-0.1180 


-0.0787 


0.0662 





* The authors are indebted to the National Research Council of Canada for a grant 
which greatly facilitated the computation of the table. 

is the largest bending moment at the center of the plate. Table 54 shows, however, 
that thie difference between M max and the largest of the values of M x and M y does not 
exceed 10 per cent of the latter values and that the general procedure described on 
page 238 is justified in case 4 as well. 

For the purpose of the design of isolated panels without continuity (Fig. 119), 
Table 54 contains the values of the largest moments ilf max acting at x = 0.1a, y = 0.16; 
for rectangular plates the direction of (7 max is practically that of the shorter span and 
for square plates that of the diagonal x = — y. For the sake of a greater security 
t hose values of M mAX may also be used in calculating continuous panels of oblong shape. 



244 



THEORY OF PLATES AND SHELLS 



—a — 



1 



1 4 
1 v 

1 y / 



3 s ^ 



2* 



<*,£ 



r^ 



Fig. 120 

Table 55. Bending Moments for Uniformly Loaded Plates in Case 5* 
v =» 0.2, / = smaller of spans a and b 





Center of plate 


Middle of fixed edge 




b/a 


M x = a 6 gZ 2 


M v = foql* 


Mxytfl* 


My = Ssql* 


Factor 




«6 


13* 


7s 


h 




0.5 
0.6 
0.7 
0.8 


0.0206 
0.0245 
0.0268 
0.0277 


0.0554 
0.0481 
0.0409 
0.0335 


-0.0783 
-0.0773 
-0.0749 
-0.0708 


-0.114 
-0.102 
-0.0907 
-0.0778 


qb* 


0.9 
1.0 


0.0274 
0.0261 
0.0294 
0.0323 


0.0271 
0.0213 
0.0204 
0.0192 


-0.0657 
-0.0600 
-0.0659 
-0.0705 


-0.0658 
-0.0547 
-0.0566 
-0.0573 




1.1 
1.2 




1.3 
1.4 
1.5 
1.6 


0.0346 
0.0364 
0.0378 
0.0390 


0.0179 
0.0166 
0.0154 
0.0143 


-0.0743 
-0.0770 
-0.0788 
-0.0803 


-0.0574 
-0.0576 
-0.0569 
-0.0568 


qa* 


1.7 
1.8 
1.9 
2.0 


0.0398 
0.0405 
0.0410 
0.0414 


0.0133 
0.0125 
0.0118 
0.0110 


-0.0815 
-0.0825 
-0.0831 
-0.0833 


-0.0567 
-0.0567 
-0.0566 
-0.0566 




50 


0.0417 0.0083 


-0.0833 


-0.0566 





* The data of this table are due substantially to F. Czerny, Bautech.-Arch., vol 11 
p. 33, W. Ernst & Sohn, Berlin, 1955. ' 

The method given in this article is still applicable if the spans, the flexural rigidities, 
or the intensity of the load differs only slightly from panel to panel of the continuous 
plate. Otherwise more exact methods should be used. 

It should be noted, however, that the application of the rigorous methods to the 
design of continuous floor slabs often leads to cumbersome calculations and that the 
accuracy thus obtained is illusory on account of many more or less indeterminable 
factors affecting the magnitude of the moments of the plate. Such factors are, for 
example, the flexibility and the torsional rigidity of the supporting beams, the restrain- 



CONTINUOUS RECTANGULAR PLATES 



245 



r 



— > 



" 8 I 

a,/} ?£_ 

J 

v//////, 



'//////A 



Fig. 121 

Table 56. Bending Moments for Uniformly Loaded Plates in Case 6 
v = 0.2, I = the smaller of spans a and b 





Center of plate 


Middle of fixed edge 




b/a 










Factor 




M x = atql 2 


My = foql 2 


M x = y 6 ql 2 


My = S 6 ql 2 






«6 


fit 


76 


56 







0.0083 


0.0417 


-0.0571 


-0.0833 




0.5 


0.0118 


0.0408 


-0.0571 


-0.0829 




0.6 


0.0150 


0.0381 


-0.0571 


-0.0793 




0.7 


0.0178 


0.0344 


-0.0569 


-0.0736 


qb* 


0.8 


0.0198 


0.0299 


-0.0559 


-0.0664 




0.9 


0.0209 


0.0252 
0.0213 


-0.0540 
-0.0513 


-0.0588 
-0.0513 




1.0 


0.0213 




1.1 


0.0248 


0.0210 


-0.0581 


-0.0538 




1.2 


0.0284 


0.0203 


-0.0639 


-0.0554 




1.3 


0.0313 


0.0193 


-0.0687 


-0.0563 




1.4 


0.0337 


0.0181 


-0.0726 


-0.0568 




1.5 


0.0358 


0.0169 


-0.0757 


-0.0570 


qa* 


1.6 


0.0372 


0.0157 


-0.0780 


-0.0571 




1.7 


0.0385 


0.0146 


-0.0799 


-0.0571 




1.8 


0.0395 


0.0136 


-0.0812 


-0.0571 




1.9 


. 0402 


0.0126 


-0.0822 


-0.0571 




2.0 


0.0408 


0.0118 


-0.0829 


-0.0571 




00 


0.0417 


0.0083 


-0.0833 


-0.0571 





ing effect of the surrounding walls, the anisotropy of the plate itself, and the inaccuracy 
in estimating the value of such constants as the Poisson ratio p. 

However, we can simplify the procedure of calculation by restricting the Fourier 
series, representing a bending moment in the plate, to its initial term or by replacing 
the actual values of moments or slopes along some support of the plate by their 
average values or, finally, by use of a moment distribution procedure. 1 

64. Bending of Plates Supported by Rows of Equidistant Columns— 
(Flat Slabs). If the dimensions of the plate are large in comparison with 

1 For such methods see C. P. Siess and N. M. Newmark, Univ. Illinois Bull 43, 
1950, where a further bibliography on the subject is given. See also the paper of 
H. M. Westergaard, Proc. Am. Concrete Inst. } vol. 22, 1926, which contains valuable 
conclusions regarding the design of continuous floor slabs. 



246 



THEORY OF PLATES AND SHELLS 



the distances a and 6 between the columns (Fig. 122) and the lateral load 
is uniformly distributed, it can be concluded that the bending in all 
panels, which are not close to the boundary of the plate, may be assumed 
to be identical, so that we can limit the problem to the bending of one 
panel only. Taking the coordinate axes parallel to the rows of columns 
and the origin at the center of a panel, we may consider this panel as a 
uniformly loaded rectangular plate with sides a and 6. From symmetry 
we conclude that the deflection surface of the plate is as shown by the 
dashed lines in Fig. 1226. The maximum deflection is at the center of 
the plate, and the deflection at the corners is zero. To simplify the 
problem we assume that the cross-sectional dimensions of the columns 
are small and can be neglected in so far as deflection and moments at 




4 

1 
1 

1 

0|CsJ 

1 
1 
1 




f-§~. 


i 

\ 

\ 
\ 
\ 
i 


n — 




p 


/ 

/ 

/ 
/ 
/ 

— n 



(b) 




Fig. 122 

the center of the plate are concerned. 1 We then have a uniformly loaded 
rectangular plate supported at the corners, and we conclude from sym- 
metry that the slope of the deflection surface in the direction of the 
normal to the boundary and the shearing force are zero at all points 
along the edges of the plate except at the corners. 2 

Proceeding as in the case of a simply supported plate (Art. 30), we 
take the total deflection w in the form 



where 



w = Wi -f- Wz 

1 384Z) \ x 6* ) 



(a) 
(b) 



1 In this simplified form the problem was discussed by several authors; see, for 
example, A. Nadai, tlber die Biegung durchlaufender Platten, Z. angew. Math. 
Mech., vol. 2, p. 1, 1922, and B. G. Galerkin, "Collected Papers," vol. 2, p. 29, Mos- 
cow, 1953. 

« The equating to zero of the twisting moment M xy along the boundary follows from 
the fact that the slope in the direction of the normal to the boundary is zero. 



CONTINUOUS RECTANGULAR PLATES 247 

represents the deflection of a uniformly loaded strip clamped at the ends 
y = ±6/2 and satisfies the differential equation (103) of the plate as well 
as the boundary conditions 

\ dX /x-dkfl/2 3X \ dX 2 dy 2 /x-±a/2 

The deflection w 2 is taken in the form of the series 

OO 

w 2 = Ao + / F m cos — — (d) 

L^ a 

m = 2,4,6, . . . 

each term of which satisfies the conditions (c). The functions Y m must 
be chosen so as to satisfy the homogeneous equation 

AAw 2 = (e) 

and so as to make w satisfy the boundary conditions at the edges 
■// = ±6/2. Equation (e) and the conditions of symmetry are satisfied 
by taking series (d) in the form 

00 

w 2 = A + / I A m cosh ^^ + B m ^^ sinh ^^ ) cos ^? (/) 
Z-/ \ a a a / a 

m = 2,4,G, . . . 

where the constants Ao, A m , and J5 m are to be determined from the 
boundary conditions along the edge y = 6/2. From the condition con- 
cerning the slope, viz., that 



\dy)v=b/2 \dy dy )y= h/2 



we readily find that 
in which, as before, 



D , tanh a m , . 

B m = —A m — — — r (g) 

otm + tanh a m 



rrnrb /X v 



Considering now the boundary condition concerning the shearing force, 
we see that on a normal section nn (Fig. 1226) of the plate infinitely 
(dose to the boundary y = 6/2, the shearing force Q y is equal to zero at 
all points except those which are close to the column, and at these points 
Q y must be infinitely large in order to transmit the finite load %qab to the 
column (Fig. 122c) along an infinitely small distance between x = a/2 — c 
and x = a/2 + c. Representing Q y by a trigonometric series which, from 



248 THEORY OF PLATES AND SHELLS 

symmetry, has the form 



oo 

q v = Co + Y, Cm cos ^r w 



m = 2,4,6, .. 

and observing that 

Q y = for < x < | - c 
and f a/2 Q ydx =-** 

Ja/2-c 4 

we find, by applying the usual method of calculation, that 

°°" ~ 25"" ~Ya 
and C m = - /" ° /2 Q„ cos — tfc; = - - (- l)-/a 

where P = ga& is the total load on one panel of the plate. Substituting 
these values of the coefficients C and C m in series (i), the required bound- 
ary condition takes the following form: 

(Q^ - D ( d * w 4. d * w ^ 

\3y 8 &c 2 dy/y^/2 



^ V / 1W/2 W7rX -P 

= / (— l) w/2 COS — 



m = 2,4,6, . . 



Substituting expression (a) for w and observing that the second term in 
parentheses vanishes, on account of the boundary condition dw/dy = 0, 
we obtain 



-KSL.--S t <-»- 



cos 



a 

m = 2,4,6, 



from which, by using expression (/), we find that 

D —5- [(A m + 3£ m ) sinh <* m + B m a m cosh «,] = - (- l)«/« (y) 

vv Co 

Solving Eqs. (0) and 0') for the constants 4 TO and £ m , we obtain 

A ro = _ -Pfl 2 /_jw/2 "» + tann gg 
2m% 3 Z) ; sinh a ro tanh a m 

B - = o 3 an (-l) m/2 --c (k) 

2m 3 ir 3 D y sinh «„» w 



CONTINUOUS RECTANGULAR PLATES 

The deflection of the plate takes the form 



249 



w 



00 



(_l)m/2 CO g 



a 



2w z D Zv w 3 sinh a m tanh a n 

m = 2,4,6, 



[tanh a m ^ sinh ^ - <«. + tanh « m ) cosh "&] 



0> 



The constant A can now be determined from the condition that the 
deflection vanishes at the corners of the plate. Hence 

(w)s=a/2,y=6/2 = 



and 



Ao 2t 



00 

qa z b V * /" a OT + tanh a m \ , . 



m = 2,4,6,... 



The deflection at any point of the plate can be calculated by using expres- 
sions (J) and (ra). The maximum deflection is evidently at the center of 
the plate, at which point we have 



(w) 



gb 4 



—O.y-0 384I) 27r 3 



00 

qa*b V 
2tt 8 D Li 



(- ! )■»/» q OT + tanha w 
3 sinh a m tanh a. 



m 



m = 2,4,6, 



00 

_ g q3& V i_ ( __ "m + tanh a OT \ 
2^D Z/ m 8 \ am tanh 2 a m / 



(n) 



m = 2,4,6,... 



Values of this deflection calculated for several values of the ratio b/a are 
given in Table 57. Values of the bending moments (Af^s-o^o and 
(M y ) x =o,y=o calculated by using formulas (101) and expression (I) for 
deflection are also given. It is seen that for & > a the maximum bend- 

Table 57. Deflections and Moments at the Center of a Panel (Fig. 122) 

v - 0.2 





qb* 
w = a — 


M m = pqb* 


My = Piqb* 


b/a 


D 








a 


P 


0i 


1 


0.00581 


0.0331 


0.0331 


1.1 


0.00487 


0.0261 


0.0352 


1.2 


0.00428 


0.0210 


0.0363 


1.3 


0.00387 


0.0175 


0.0375 


1.4 


0.00358 


0.0149 


0.0384 


1.5 


0.00337 


0.0131 


0.0387 


2.0 


0.00292 


0.0092 


0.0411 


00 


0.00260 


0.0083 


0.0417 



250 



THEORY OF PLATES AND SHELLS 



ing moment at the center of the plate does not differ much from the 
moment at the middle of a uniformly loaded strip of length b clamped 
at the ends. 

Concentrated reactions are acting at the points of support of the plate, and the 
moments calculated from expression (I) become infinitely large. We can, however, 
assume the reactive forces to be distributed uniformly over the area of a circle repre- 
senting the cross section of the column. The bending moments arising at the center 
of the supporting area remain finite in such a case and can be calculated by a pro- 
cedure similar to that used in the case of rectangular plates and described on page 147. 
With reference to Fig. 122, the result can be expressed by the formulas 1 



{M s ) x . 



a/2,y-b/2 



= Mn — 



i + o- 



qb 2 

(M y ) x -an,v-bn = Mo + — r 



In these expressions 



n=l a J 

L~4 sinh* 



(o) 



— 7 + (1 - ") -T 

nvo wo 



Mt 



_ 2°^ r 



, a + ") log 

4ir l J 6 2irc(l - g*)*(l - g<) 2 



+ 1 



] 



q = e~ Tbla j and c denotes the radius of the circle, supposed to be small compared 
with spans a and b of the panel. Carrying out the required calculations, we can 
reduce Eqs. (o) to the form 



(Af x ) a: « a /2, 1/ „b/2 = — 



qab 



(M, 



(1 +„)log^- (a+M 

qab [ a I 

,)*~o/2.i,-&/2 = - — (1 + v) log (p + av) \ 



(P) 



in which a and p are coefficients given for several values of the ratio b/a in Table 58. 

Table 58. Values of Coefficients a and in Eqs. (p) for Moments 

on Support 



b/a 


1 


1.1 


1.2 


1.3 


1.4 


1.5 


2.0 


Ot 


0.811 
0.811 


0.822 
0.698 


0.829 
0.588 


0.833 
0.481 


0.835 
0.374 


0.836 
0.268 


0.838 
-0.256 



The bending moments corresponding to the centers of columns of rec- 
tangular cross section also can be calculated by assuming that the reac- 
tions are uniformly distributed over the rectangles, shown shaded in Fig. 



1 Given by A. Nddai in his book "Elastische Flatten," p. 154, Berlin, 1925. 



CONTINUOUS RECTANGULAR PLATES 



251 



Table 59. Bending Moments and Largest Shear Force of a Square 
Panel of a Uniformly Loaded Plate (Fig. 123) 

v = 0.2 



n/a = k 


(Jl/)x-y~a/2 = 0$a 2 


(M) x - v -o = 0iga 2 


(JWx)x-a/2,y-0 = 02tfa 2 


(My)x-a/2,y-0 * 0300* 


Qmax — 7(7« 

j 




P 


0i 


02 


03 


7 





— 00 


0.0331 


-0.0185 


0.0512 


1 

00 


0.1 


-0.196 


0.0329 


-0.0182 


0.0508 


2.73 


0.2 


-0.131 


0.0321 


-0.0178 


0.0489 




0.3 


-0.0933 


0.0308 


-0.0170 


0.0458 


0.842 


0.4 


-0.0678 


0.0289 


-0.0158 


0.0415 




0.5 


-0.0487 


0.0265 


-0.0140 


0.0361 


0.419 



123, that represent the cross sections of the columns. 1 In the case of 
square panels and square columns we have u/a = v/b = /c, and the 
moments at the centers of the columns and at the centers of the panels 
are given by the following formulas : 

(1 + v)qa* [(1 - fe)(2 - k) 
12 



(M z ) x =r ys =a/2 = (My) x -y-a/2 — "" 



t 



+ 



1 V 2 . . mirfc , m7r(2 - fc) - ,1 

fro / 3 • u sinh — o" cosh o sin m ™ 

3 fc 2 Z_/ ra 3 sinh m7r 2 2 J 



TfK' /_/ m 3 sinh mir 

m—l 



(1 + v)qa 2 



L 12 ^ 7T 3 fc 2 Zy ^ ' 

771= 1 



sinh ra7r/c sin m^rfc 
ra 3 sinh W7r 



] 



(«) 



(r) 



The values of these moments, to- 
gether with values of moments at 
half a distance between columns, ob- 
tained from the same solution and 
calculated for various values of k and 
for v = 0.2, are given in Table 59. 

It is seen that the moments at the 
columns are much larger than the 
moments at the panel center and that 
their magnitude depends very much 
on the cross-sectional dimensions of 
the columns. The moments at the 
panel center remain practically con- FlG * 123 

stant for ratios up to k = 0.2. Hence the previous solution, obtained on 

1 This case was investigated by S. Woinowsky-Krieger; see Z. angew. Math. Mech., 
vol. 14, p. 13, 1934. See also the papers by V. Lewe, Bauingenieur, vol. 1, p. 631, 
1920, and by K. Frey, Bauingenieur, vol. 7, p. 21, 1926. 




252 



THEORY OP PLATES AND SHELLS 



the assumption that the reactions are concentrated at the panel corners, 
is sufficiently accurate for the central portion of the panel. 

An approximate calculation of moments given by Eq. (q) in the form of a series can 
also be made by means of expressions (p). Using for this purpose Eq. (c), Art. 37, 
we substitute 



u 



c = 



vs 



6 T/4-l = Q m57u 



i.e., the radius of a circle equivalent to the given square area u by u, in Eqs. (p). In 
the particular case of square panels numerical results obtained in this manner are but 
slightly different from those given in the second column of Table 59. 

The shearing forces have their maximum value at the middle of the 
sides of the columns, at points m in Fig. 123. This value, for the case of 
square panels, depends on the value of the ratio k and can be represented 
by the formula Q = yqa 2 . Several numerical values of the factor y are 
given in Table 59. It is interesting to note that there is a difference of 
only about 10 per cent between these values and the average values 
obtained by dividing the total column load qa 2 (l — fc 2 ) by the perimeter 

4/ba of the cross section of the column. 
Uniform loading of the entire plate 
gives the most unfavorable condition 
at the columns. To get the maxi- 
mum bending moment at the center 
of a panel, the load must be distri- 
buted as shown by the shaded areas 
in Fig. 124a. The solution for this 
case is readily obtained by combining 
i the uniform load distribution of in- 
iiUUKUHlUUlQ Ij tensity q/2 shown in Fig. 1246 with 

the load q/2 alternating in sign in 
consecutive spans shown in Fig. 124c. 
The deflection surface for the latter 
case is evidently the same as that 
for a uniformly loaded strip of length 
a simply supported at the ends. 
Taking, as an example, the case of square panels and using the values in 
Table 57, we find for the center of a panel (Fig. 124a) : 




(b) 



if 



TTTTT 



HtHH 

<c) 



Fig. 124 



fTTTTT 



i 



Br 



(w)*-»-o 

(Afx)*=j/=0 



1,. 0.00581 £ + 3 -y£ = 0.00942 £ 



^ q • 0.0331a 2 + ^ qa 



2 _ 



0.0791ga 2 



^ q • 0.0331a 2 + ^| ga 2 = 0.0291ga 2 



CONTINUOUS RECTANGULAR PLATES 



253 




Fig. 125 
Taking the case of a 



From Table 59 we conclude, furthermore, that 

(ilf x )_ .y-6/2 = iQ ' 0.0512a 2 + T \qa> = 0.0881ga 2 

The foregoing results are obtained in assuming that the plate is free to 
rotate at the points of support. Usually the columns are in rigid con- 
nection with the plate, and, in the case of the load distribution shown in 
Fig. 124, they produce not only vertical reactions but also couples with a 
restraining effect of those couples on the bending of the panels. A frame 
analysis extended on the flat slab and the columns as a joint structure 
therefore becomes necessary in order to obtain 
more accurate values of bending moments under 
alternate load. 1 

The case in which one panel is uniformly 
loaded while the four adjacent panels are not 
loaded is obtained by superposing on a uniform 
load q/2 the load g/2, the sign of which alter- 
nates as shown in Fig. 125. In this latter case 
each panel is in the same condition as a simply 
supported plate, and all necessary information 
regarding bending can be taken from Table 8. 
square panel, we find for the center of a panel that 

(u>Wo = \z ' o- 00581 j) + 5 * ' - 00406 % = 000494 % 

(M,)„o = (M y Wo = \q ' 0.0331a* + \ q • 0.0479 ' {| a 2 = 0.0387ga* 

The case of an infinitely large slab subjected to equal concentrated 
loads centrally applied in all panels can be handled substantially in the 
same manner as in the preceding case, i.e., by using the double periodicity 
in the deflections of the plate. 2 

The problem of bending of a uniformly loaded flat slab with skew 
panels has also been discussed. 3 

55. Flat Slab Having Nine Panels and Slab with Two Edges Free. 
So far, an infinite extension of the slab has always been assumed. Now 
let us consider a plate simply supported by exterior walls, forming the 
square boundary of the plate, together with four intermediate columns 
(Fig. 126). From symmetry we conclude that a uniformly distributed 

1 The procedure to be used is discussed in several publications; see, for instance, 
H. Marcus, "Die Theorie elastischer Gewebe," p. 310, Berlin, 1932. 

2 This problem was discussed by V. Lewe in his book "Pilzdecken und andere 
tragerlose Eisenbetonplatten," Berlin, 1926, and also by P. Pozzati, Riv. math. Univ. 
Parma, vol. 2, p. 123, 1951. 

3 See V. I. Blokh, Doklady Akad. Nauh S.S.S.R., n. s., vol. 73, p. 45, 1950. 



254 



THEORY OF PLATES AND SHELLS 




load of intensity q produces equal 
column reactions R, which we may 
consider as redundant in the given 
statically indeterminate structure. 
Removing all columns, we obtain a 
simply supported square plate carry- 
ing merely the given load q. The 
deflections w produced by this load 
at the center of the columns can 
easily be calculated by means of the 
theory given in Chap. 5. Next, re- 
moving the load q and distributing a 
load R = 1 (acting downward) uni- 
formly over each area u by u, we 
obtain some new deflections W\ at the 
same points x = +a/2, y = ±a/2 
as before. From the condition that in 
the actual case these points do not de- 
flect, we conclude that w — Rwi = 0, 
which yields R = w /wi. Now it 
remains only to combine the effect 
of the uniform load q with the effect 
of four known reactions on the bend- 
ing moments of the square plate of 
the size 3a by 3a. 

In the case of a partial loading, 
such as shown in Fig. 1266 and c, we 
have to superpose one-half of the 
moments previously obtained on the 
moments of a simply supported plate 
with the area a by 3a, carrying a 
uniformly distributed load ±q/2. 
Calculations of this kind carried out 
by Marcus 1 led to the values of bend- 
ing moments given in Table 60. The 
reaction of a column is R = 1.196ga 2 
in this case. The bending of an in- 
finite plate which is supported not 
only along both its parallel sides 



126 



also Lewe, op. cit. 



1 "Die Theorie elastischer Gewebe"; see 
The case of a square plate with one intermediate support was 
discussed by N. J. Nielsen, "Bestemmelse af Spaendinger I Plader," p. 217, Copen- 
hagen, 1920. 



CONTINUOUS RECTANGULAR PLATES 



255 



Table 60. Coefficients p for Calculation of Bending Moments M = Pqa 2 
of a Simply Supported Square Plate with Four Intermediate 

Columns (Fig. 126) 
u/a = 0.25, v - 0.2 



Point 


X 

a 


y 

a 


Load a 


Load b 


Load c 


M x 


My 


M x 


My 


M x 


My 


1 

2 
'A 
4 
5 

6 

7 
8 
9 




5 
1.0 

0.5 

1.0 


0.5 
1.0 








0.5 

0.5 

0.5 
1.0 
1.0 
1.0 


0.021 

-0.040 

0.069 

0.038 

-0.140 

0.074 

0.025 

-0.004 

0.053 


0.021 

0.038 

0.025 

-0.040 

-0.140 

-0.004 
0.069 
0.074 
0.053 


-0.048 
-0.020 
0.093 
-0.036 
-0.070 

0.092 
-0.028 
-0.002 

0.066 


-0.004 

0.019 

0.027 

-0.036 

-0.070 

0.014 
0.017 
0.037 
0.044 


0.069 
-0.020 
-0.024 

0.074 
-0.070 

-0.018 

0.052 

-0.002 

-0.013 


0.025 

0.019 

-0.002 

-0.004 

-0.070 

-0.018 
0.052 
0.037 
0.009 



but also by one or several rows of equidistant columns 1 can be discussed 
in a similar manner. 

The case of bending of a long rectangular plate supported only by the 
two parallel rows of equidistant columns (Fig. 127) can also be solved 
without any difficulty for several types of loading. We begin with the case 
in which the plate is bent by the moments M y represented by the series 



(My)y= ±b/ 2 = M0i 



00 

I 



E m COS 



TYlTTX 

a 



(a) 



m = 2,4,6, ... 

Since there is no lateral load, the deflection surface of the plate can be 
taken in the form of the series 

4 



w = Ao+ Ally 



+ 



2 ( 



. . miry , „ miry • i miryx rmrx ... 

A m cosh — - - + B m —± sinh — ^- J cos —7- (6) 



a J 



a a a / a 

m = 2,4,6, 

the coefficients of which are to be determined from the following boundary- 
conditions : 






= M + 



=±6/2 



I 



m = 2,4,0, 



+ (2 - v) 



d z W 



E m cos 



= 



rrnrx 
a 



(c) 



dy 3 ' v " '' dy dx 2 ] y=±b ,2 
ir rhis problem has been considered by K. Grein, "Pilzdecken," Berlin, 1948. 



256 THEORY OF PLATES AND SHELLS 

and from the condition that the deflection vanishes at the columns. 
Substituting series (6) in Eqs. (c), we find that 

Al ~ 2D 

A = - a2Em (1 + v) sinh a m - (1 - y)g m cosh a m 

w*m*D (3 + „)(1 - v) sinh a m cosh a ro - a m (l - pj~* W 
t> _ a?Em sinh a m 



ir 2 m 2 D (3 + v) sinh a m cosh a m - a m (l - v) 

Combining this solution with solution if), Art. 54, we can investigate the 
bending of the plate shown in Fig. 127a under the action of a uniformly 




(b) 
Fig. 127 

distributed load. For this purpose we calculate the bending moments 
M v from expression (I) by using formula (101) and obtain 



(Afy) v -±6/2 



_ £&2 

12 

qab 
2^ 

to = 2,4,6, 



00 

V (~ 1 ) m/2 {A+JL _ Ml - r) 1 mxa; 
Z/ m [tanh a m sinh 2 a m J cos a (e; 



Equating this moment to the moment (a) taken with the negative sign, 
we obtain the values of M and E n which are to be substituted in Eqs. 
(d) for the constants A h A m , and B m in expression (6). Adding expres- 
sion (b) with these values of the constants to expression (I), Art. 54, w© 
obtain the desired solution for the uniformly loaded plate shown in Fig 
127a. 

Combining this solution with that for a uniformly loaded and simply 
supported strip of length b which is given by the equation 



w 



--akS-'XS*-.-) 



we obtain the solution for the case in which the plate is bent by the load 
uniformly distributed along the edges of the plate as shown in Fig. 1276. 



CONTINUOUS RECTANGULAR PLATES 



257 



66. Effect of a Rigid Connection with Column on Moments of the Flat Slab. In 

discussing the bending of a flat slab it has always been assumed that the column 
reactions are concentrated at some points or distributed uniformly over some areas 
corresponding to the cross section of the columns or their capitals. As a rule, however, 
concrete slabs are rigidly connected with the columns, as shown in Fig. 128. 

In discussing moments at such rigid joints, let us begin with the case of a circular 
column and let c be the radius of its cross section. The calculation o* bending 













< 0.22 a— —> 








; 


' 1 


' i 


' i 


' i 


p i- if 


1, M 1 


r , 


■ • 


- < 


W//////A 


WW////, 


* 




4 


I • 








' J ' i{ a qa 2 
ire* 


r 




* 










<—2 


>c— > 









Fig. 128 

moments using expression (I) in Art. 54 shows 1 that, in the case of a square panel 
(a = b) and small values of c/a, the bending moments in the radial direction practically 
vanish along a circle of radius e = 0.22a (Fig. 122a). Thus the portion of the plate 
around the column and inside such a circle is in the state of an annular plate simply 
supported along the circle r = 0.22a and clamped along the circle r = -• c, with a 
transverse displacement of one circle with respect to the other. Hence the maximum 



,* Middle line of panel 



ll 

O i 
°* 1 




•*- 


c = 


0.1a 

<— 


i 

1 

1 

d|c\j 

1 


M\ 




l 


& 




I 


-XD 1 
1 


K 


t 

i 
i 
i 

o|c\J 

! 
1 












1 




< 




'2" * 


— > 


--I--J 








y 










F 


IG. 


129 





^-Middle line of panel 




Fig. 130 

bending stress around the column can be obtained by using formulas (75), previously 
derived for circular plates (see page 61), and combining cases 3 and 8 in Fig. 36. 

A more elaborate discussion of the same problem is due to F. Tolke. 2 Numerical 
results obtained by F. Tolke for a square panel and c/a = 0.1 (Fig. 129) are given in 
Table 61, together with values of bending moments calculated for the same case on the 

1 Such calculations were made by A. Nadai; see his book "Elastische Platten," p. 
156, Berlin, 1925. 

2 F. Tolke, Ingr.-Arch., vol. 5, p. 187, 1934. 



258 



THEORY OF PLATES AND SHELLS 



basis of the customary theory. It is seen that a rigid connection between slab and 
column tends to increase numerically the moments on support and to reduce the posi- 
tive moments of the slab. 

Table 61. Coefficients for Calculation of Bending Moments M = fiqa 2 
of a Uniformly Loaded Square Panel of a Flat Slab 

v = 0.2 





Location 


Circular column 
(Fig. 129) 


Square column 
(Fig. 130) 


Bending 
moment 


Rigid 

connection 

with 

column 


Customary 
theory 


Rigid 

connection 

with 

column 


Customary 
theory 


M X = My 

M x 

My 
M x = My 

M x 

Mr 
Mr 


x = a/2, y = a/2 

x = a/2, y = 

x = a/2, y = 

x = 0, y = 

x = u/2, y = 

x = u/2, y = u/2 

r = c 


0.0292 

0.0399 

-0.0161 

-0.1682 


0.0323 
0.0494 
-0.0179 
-0.143 

-0.0629 


0.0264 

0.0348 

-0.0146 

-0.0626 

— 00 


0.0321 
0.0487 
-0.0178 
-0.131 
-0.0803 
-0.0480 



The same table also gives moments for a flat slab rigidly connected with a column 
of a square cross section 1 (Fig. 130). The infinitely large stresses occurring at the 
corners of columns in this case are of a highly localized character. Practically, they 
are limited by a cracking of concrete in tension and a local yielding of the steel 
reinforcement. 

From this discussion we may conclude that (1) the actual values of bending moments 
of a flat slab at the columns generally lie between the values given in Table 61 for the 
rigid connection and those given by the usual theory, and (2) circular columns secure 
a more uniform distribution of clamping moments than columns with a square-shaped 
supporting area. 2 

1 See S. Woinowsky-Krieger, J. Appl. Mechanics, vol. 21, p. 263, 1954. 

2 See T. Haas, "Conception et calcul des planchets k dalles champignon," Paris, 
1950. The distribution of stresses in a flat slab has been investigated experimentally 
by M. Ro§ and A. Eichinger, Proc. Congr. Concrete and Reinforced Concrete, Li6ge ; 
1930; by R. Caminade and R. L'Hermite, Ann. inst. tech. bdtiment et trav. publ., 
February, 1936; and more recently by J. G. Hageman, Ingenieur, vol. 65, June, 1953. 



CHAPTER 8 



PLATES ON ELASTIC FOUNDATION 



57. Bending Symmetrical with Respect to a Center. A laterally 
loaded plate may rest on an elastic foundation, as in the case of a con- 
crete road, an airport runway, or a mat. We begin the discussion of such 
problems with the simplest assumption that the intensity of the reaction 
of the subgrade is proportional to the deflections w of the plate. This 
intensity is then given by the expression kw. The constant fc, expressed 
in pounds per square inch per inch of deflection, is called the modulus of 
the foundation. The numerical value of the modulus depends largely 
on the properties of the subgrade; in the case of a pavement slab or a 
mat of greater extension this value may be estimated by means of the 
diagram in Table 62. l 

Table 62. Values of the Modulus of Subgrade 



Modulus V in Ib/sq in./in. 



100 



150 



2Q0 



250 



500 



800 



General soil rating as subgrade, subbase or base 



Very poor subgrade 



Poor 
subgrade 



Fair to good subgrade 



G- Gravel P - Poorly groded ' 

S - Sand L - Low to med. compressibility 

M-Mo^'very fine sand, silt H - High compressibility 

C-Clay 

F - Fines, material 

less than 0.1mm 
0- Organic 
W-Wellgroded 



Excellent 
subgrade 



Good subbase 



Good 
base 



Best 
base 



GW 



GC 



GP 



GF 



sw 



sc 



SP 



SF 



CH 



ML 



OH 



CL 



OL 



MH 



Let us begin with the case of a circular plate in which the load is dis- 
tributed symmetrically with respect to the center. In using Eq. (58), 

1 Based on Casagrande's soil classification. The table should not be regarded as a 
substitute for plate bearing tests. For further information see Trans. Am. Soc. Civ, 
Engrs., vol. 113, p. 901, 1948. See also K. Terzaghi, Geotechnique, vol. 5, p. 297, 1955 
(Harvard Soil Mechanics Series, no. 51). 

259 



260 THEORY OF PLATES AND SHELLS 

we add the load —kw, due to the reaction of the subgrade, to the given 
lateral load q. Thus we arrive at the following differential equation for 
the bent plate: 

(*. + 1 ± \(*» + I dw\ _ q - kw 

V*r 2 ^ r drj \dr* ^ r dr ) D~ {1 ™> 

In the particular case of a plate loaded at the center with a load P* 
a is equal to zero over the entire surface of the plate except at the center. 
By introducing the notation 

k - 1 r, 

Eq. (178) becomes 

(d* 1 d\ (d*w ,ldw\, n 

1 \dF> + -rTr)\W* + -rdF) + W - ° W 

Since k is measured in pounds per cubic inch and D in pound-inches, the 
quantity I has the dimension of length. To simplify our further dis- 
cussion it is advantageous to introduce dimensionless quantities by using 
the following notations: 

w r 

J - Z j - X ( C ) 

Then Eq. (6) becomes 

/d» 1 d\/dh,ldz\, n 

\d* + xte)\w + xdx-) + Z = ° W 

Using the symbol A for 

^ H 

dx 2 x dx 
we then write 

AAs + z = (e) 

This is a linear differential equation of the fourth order, the general solu- 
tion of which can be represented in the following form : 

z = AxXi(x) + ^ 2 X 2 (o;) + A z X z (x) + A A X A (x) (/) 

where A h . . . , A A are constants of integration and the functions 
Xi, . . . , X A are four independent solutions of Eq. (e). 

We shall now try to find a solution of Eq. (e) in the form of a power 

* This problem was discussed by H. Hertz, Wiedemann's Ann. Phys. u. Chem., vol. 
22, p. 449, 1884; see also his " Gesammelte Werke," vol. 1, p. 288, 1895, and A. Foppl, 
" Vorlesungen liber technische Mechanik, ,, vol. 5, p. 103, 1922. It is worth noting that 
Hertz's investigation deals with the problem of a floating plate rather than with that 
of a piate on an elastic foundation. Thus, in this case the assumption regarding the 
constancy of k is fulfilled, k being the unit weight of the liquid. 



PLATES ON ELASTIC FOUNDATION 261 

series. Let a n x n be a term of this series. Then, by differentiation, we 
find 

A(a n x n ) = n{n — l)a n x n ~ 2 + na n x n - 2 = n 2 a n x n - 2 
and AA(a n x n ) = n 2 (n - 2) 2 a n x n " 4 

To satisfy Eq. (e) it is necessary that each term a n x n in the series have a 
corresponding term a n _4^ n ~ 4 such that 

n 2 (n - 2) 2 a n x n ~ A + a n .a n ^ = (g) 

Following this condition, all terms cancel when the series is substituted in 
Eq. (e) ; hence the series, if it is a convergent one, represents a particular 
solution of the equation. From Eq. (g) it follows that 

Ctn—4 



a n = — 



(h) 



n 2 (n - 2) 2 
Observing also that 

AA(a ) = and AA(a 2 x 2 ) = (t) 

we can conclude that there are two series satisfying Eq. (e), viz., 



Xi(x) = 1 - ^o-7T2 + 



x A , a: 8 



2 2 . 42 1 2 2 • 4 2 • 6 2 • 8 2 



x 12 



+ 



2 2 . 42 . 6 2 . g2 • 102 - 12 2 

and (i) 



a; 6 , x 10 



-A 2.(X) — X 42 . g2 "" 42 . Q2 . g2 . JO 2 



a; 14 



42 . 6 2 . g 2 . 1Q 2 . 122 • 14' 



+ 



It may be seen from the notations (c) that for small values of the dis- 
tance r, that is, for points that are close to the point of application of the 
load P, the quantity x is small, and series (j) are rapidly convergent. 
It may be seen also that the consecutive derivatives of series (j) remain 
finite at the point of application of the load (x = 0). This indicates that 
these series alone are not sufficient to represent the stress conditions at 
the point of application of the load where, as we know from previously 
discussed cases, the bending moments become infinitely large. 

For this reason the particular solution X 3 of Eq. (e) will be taken in 
the following form: 

X 3 = Xi log x + F z (x) (k) 

in which F z {x) is a function of x which can again be represented by a 
power series. By differentiation we find 

AAX 3 = ^ ^~ + log x AAXa + AAF-Ax) 



262 THEORY OF PLATES AND SHELLS 

and substituting X z for z in Eq. (e), we obtain 

| ^ + log x(AAXa + X x ) + AAF 3 (x) + F,(x) = 

Since Xi satisfies Eq. (e) and is represented by the first of the series (j) 
we obtain the following equation for determining F$(x) : 



a a e. / n , r? / n 4 d 3 Xi .( 2-3-4 

AAF s (a;) + F 3 (a:) = ---^=-4^- -^-^ 



x dx 



6 • 7 • 8 • x 4 10 ; 11 • 12 • x s \ 

+ 2 2 • 4 2 • 6 2 • 8 2 2 2 • 4 2 • 6 2 • 8 2 • 10 2 • 12 2 + ' ' / ^ ' 

Taking F 3 (x) in the form of the series 

F 3 (x) = biX* + bsx* + b 12 x n + • • • (m) 

and substituting this series in Eq. (I), we determine the coefficients hi, 
b», &12, ... so that the resulting equation will be satisfied. Observing 
that 

AA(b A x 4 ) = 4 2 • 2 2 • 6 4 

we find, by equating to zero the sum of the terms that do not contain x, 
that 

2-3-4 



4 2 • 2 2 • bi = 4 
or 64 = 



22.42 
2 • 3 • 4 2 3 



2 4 • 4 4 128 
Equating to zero the sum of the terms containing x 4 , we find 

h = 25 

08 1,769,472 

In general, we find 

b = r_i)»/4-i 2 L 1 n ( n ~ !)(* ~ 2 ) 1 

n ^ ; n 2 (n - 2) 2 [ " _4 ^ 2 2 • 4 2 • 6 2 • • • n 2 J 

Thus the third particular solution of Eq. (e) is 

3 25 

X * = X * l °Z* + 128^" 1,769,472 X * + ' ' ' < W > 

The fourth particular integral X4 of Eq. (e) is obtained in a similar 
manner by taking 

Xi= X 2 log x + F t (x) = X 2 log x + 4 • i^-^ x« 



10 ; 



44. 6 4 

1 / 4-5-6 10-9-8 \ } 

•8 2 V ^-6 4 4 2 -6 2 • • • IV 1 + w 



PLATES ON ELASTIC FOUNDATION 263 

By substituting the particular solutions (j), (n), and (o) in expression 
(/) we obtain the general solution of Eq. (e) in the following form: 

= a (\ ~ x * -i — A 

3 3 \ 2 2 • 4 2 2 2 • 4 2 • 6 2 • 8 2 ' / 

-r a 2 yx 4 2 . 6 2 -r 42 . 6 2 . g2 . 10 2 J 

_i_ a f/i xA . x * \i 4. a 4 

I \ ^^ 2 2 • 4 2 • 6 2 • 8 2 / g X ^ 128 X 

25 8 ^ 1 , , 17 2 x& . * ic 

x s + . . - 4. Ai \[x 2 - 77-77- + 



•) 



•1 + ,4 4 |Yz 2 - 



1,769,472 ^ I ' ' L\ 42 ' 62 4 2 • 6 2 • 8 2 • 10 2 



. 5 6 1,540 • 10- 4 10 , \ , . 

logx + M56 g 442,368 x + ' ' J (p) 



It remains now to determine in each particular case the constants of inte- 
gration Ai, . . . , A 4 so as to satisfy the boundary conditions. 

Let us consider the case in which the edge of a circular plate of radius a 
is entirely free. Making use of expression (52) for the radial moments 
and expression (55) for the radial shear force Q r , we write the boundary 
conditions as 



/d 2 w l_dw\ ___ ft 

\dr 2 r dr ) r =a 

i_ /dhv 1 dw\ = Q 

dr \dr 2 r dr / r=a 



d (d 2 w 



In addition to these two conditions we have two more conditions that 
hold at the center of the plate; viz., the deflection at the center of the 
plate must be finite, and the sum of the shearing forces distributed over 
the lateral surface of an infinitesimal circular cylinder cut out of the plate 
at its center must balance the concentrated force P. From the first of 
these two conditions it follows that the constant A 3 in the general solu- 
tion (p) vanishes. The second condition gives 

(/ o 2 ' Q r r de) r=( + P = (r) 

or, by using notation (a), 



dr \dr 2 r dr 



t 



where e is the radius of the infinitesimal cylinder. Substituting Iz for w 
in this equation and using for z expression (p), we find that for an infinitely 
small value of x equal to e/l the equation reduces to 

-/cZ 4 ^27re + P - 
U 



264 THEORY OF PLATES AND SHELLS 

from which 

Ai - SOT «> 

Having the values of the constants A z and A A , the remaining two con- 
stants yli and A 2 can be found from Eqs. (q). For given dimensions of 
the plate and given moduli of the plate and of the foundation these equa- 
tions furnish two linear equations in A\ and A 2 . 

Let us take, as an example, a plate of radius a = 5 in. and of such rigidity that 

Z = y| = 5in. 
We apply at the center a load P such that 

8wkl* 

Using this value of A K and substituting Iz for w, we find, by using expression (p) and 
taking x = a/l = 1, that Eqs. (g) give 

0.5004i + 0.250A 2 « 4.062^4 = 4.062 • 102 • 10~ 6 
0.687^! - 8.483^l 2 = 11.09A 4 « 11.09 • 102 • 10-* 

These equations give 

ili - 86 • 10~ 4 A 2 - -64 • 10-* 

Substituting these values in expression (p) and retaining only the terms that contain 
x to a power not larger than the fourth, we obtain the following expression for the 
deflection: 

w - Iz - 5 86 • 10-M 1 - grhs) ~ 64 ' 10 ~* x2 + 102 * 10 "" 6a;2 lo S * 
The deflection at the center (x = 0) is then 

ww - 43 • 10-* in. 
and the deflection at the boundary (x = 1) is 

w m in = 39.1 • 10~ 3 in. 

The difference of these deflections is comparatively small, and the pressure distributitfn 
over the foundation differs only slightly from a uniform distribution. 

If we take the radius of the plate two times larger (a = 10 in.) and retain the 
previous values for the rigidities D and k, x becomes equal to 2 at the boundary, and 
Eqs. (q) reduce to 

0.8264i + 1.980^2 - 1.208^4 
2.665 A x - 5.7454 2 - 16.37A 4 
These equations give 

Ai - 3.93A 4 - 400 • 10~ 5 A 2 - - 1.034 « - -105 • 10~* {u) 



PLATES ON ELASTIC FOUNDATION 265 

The deflection is obtained from expression (p) as 

«, = h - 5 J 4 00 • 10- (l - ^j - 105 • 10- (z* - ^J 

+ 102-10-[log,(^- i ^ i )+^- 6 ^]J 

The deflections at the center and at the boundary of the plate are, respectively, 

wWx = 2.10- 2 in. and w m \n = 0.88 • 10~ 2 in. 

It is thus seen that, if the radius of the plate is twice as large as the quantity Z, the 
distribution of pressure over the foundation is already far from uniform. The applica- 
tion of the strain energy method to the problem of bending of a plate on elastic sub- 
grade will be shown in Art. 80. 

58. Application of Bessel Functions to the Problem of the Circular Plate. The 
general solution (/) of Eq. (e) in the preceding article can also be represented in terms 
of Bessel functions. To this end we introduce into Eq. (e) a new variable £ = x V i; 
thus we arrive at the equation 

A'A'z -2=0 (a) 



in which the symbol A' stands for 



di 2 + idl 



Now Eq. (a) is equivalent to equation 

A'(A'z + z) - (A'z + z) = (6) 

and also to 

A' (A'z - z) + (A'z - z) = (c) 

Hence Eq. (a) is satisfied by the solutions of the Bessel differential equation 

d 2 z 1 & , , x 

A 's + z „ _ + - - + z = (d) 

d? £ di 

as well as by the solutions of the equation 

d 2 z 1 dz , x 

A'z - z = — ; + - — - z = (e) 

de i dZ 

which is transformable into Eq. (d) by substituting £i for £. Thus the combined solu- 
tion of Eqs. (d) and (e) can be written as 

z = Bi/ (a; Vi) + B 2 Io(xi V~i) + B 3 X (a; V~i) + B 4 K (xi V'i) (/) 

I and jff being Bessel functions of the first and second kind, respectively, and of 
imaginary argument, whereas B h B 2y . . . are arbitrary constants. The argument 
x being real, all functions contained in Eq. (/) appear in a complex form. To single 
out the real part of the solution, it is convenient to introduce four other functions, first 
used by Lord Kelvin and defined by the relations 1 

h(x V±i) = ber x ± i bei x 
K (x V ±z) = ker x ± i kei x 

1 See, for instance, G. N. Watson, "Theory of Bessel Functions/' p. 81, Cambridge, 
1948. 



266 THEORY OF PLATES AND SHELLS 

Setting, furthermore, 

Bi + B 2 = Ci/l B x - B 2 = -Cti/l 
B z + B 4 = C 4 /l B z - B 4 - -CWZ 

where the new constants Ci, C 2 , . . . are real, we obtain the following expression for 
the deflections of the plate: 

w = Ci ber x + C 2 bei x + C* kei x + C 4 ker x (h) 

All functions herein contained are tabulated functions, 1 real for real values of the 
argumsnt. 

For small values of the argument we have 

ber x = 1 - x 4 /64 + - • • 

bei x = x 2 /4 - z 6 /2,304 + . . . 

ker x = - log x + log 2 - y + <n-x 2 /16 + • • • ^ 

kei x = -(x 2 /4) log x - tt/4 + (1 + log 2 - 7 )*74 + • • • 

in which y = 0.5772157 • • • is Euler's constant and log 2 - y = 0.11593 • • • 
For large values of the argument the following asymptotic expressions hold: 



/ A 

= COS ( <r I 



V"2" 




(J) 



ber a; 

bei x 

ker x 

kei # ~ — 

'2z/* 

in which <r = x/ v2. 

The general solution (h) can be used for the analysis of any symmetrical bending of 
a circular plate, with or without a hole, resting on an elastic foundation. The four 
constants C, corresponding in the most general case to four boundary conditions, must 
be determined in each particular case. 2 

2 See "Tables of Bessel Functions J (z) and J x (z) for Complex Arguments," 
Columbia University Press, New York, 1943, and " Tables of Bessel Functions 
Y (z) and Y x (z) for Complex Arguments," Columbia University Press, New York, 
1950. We have 

ber x = Re [J (xe iv '*)] bei x = - Im [J Q (xe iir '*)] 

ker x » - ^ Re [r (a* iir/4 )] - £ Im [Jote'" 4 )] 
^ 2 

kei x = ^ Im [yoCa**'" 4 )] - £ Re [Jotee*"*)] 
^ 2 

2 Many particular solutions of this problem are given by F. Schleicher in his 
book "Kreisplatten auf elastischer Unterlage," Berlin, 1926, which also contains 
tables of functions Z x {x) = ber x, Z 2 (x) = -bei x } Z z (x) = -(2/tt) koi x, and 
Z A (x) = — (2/tt) ker x as well as the first derivatives of those functions. An abbrevi- 
ated table of the functions Z and their first derivatives is given in Art. 118, where 
they are denoted by the symbol ^. 



PLATES ON ELASTIC FOUNDATION 267 

We shall confine ourselves to the case of an infinitely extended plate carrying & 
single load P at the point x = 0. Now, from the four functions forming solution (A), 
the first two functions increase indefinitely with increasing argument in accordance 
with Eqs. (j) ; and the function ker x becomes infinitely large at the origin, as we can 
conclude from Eqs. (i). Accordingly, setting Ci = C 2 = C 4 = 0, solution (h) is 
reduced to 

w = Cz kei x (k) 

In order to determine the constant C 3 , we calculate, by means of Eqs. (i), the shearing 
force [see Eqs. (193)] 

D d_ /dhv 1 dw\ _ C3D /l _ « \ 

Qr ~ ~ Pfa\I^ + xdx) ~ P \x 8 / 

As x decreases, the value of Q r tends to C*D/l*x = C s D/l 2 r. On the other hand, upon 
distributing the load P uniformly over the circumference with radius r, we have 
Q r = —P/2Trr. Equating both expressions obtained for Q r , we have 

PI 2 

Substitution of C 3 into Eq. (k) yields, finally, the complete solution of Hertz's problem 
in the form 

PI 2 



w = 



2wD 



kei x (179) 



wD 

and the corresponding reaction of the subgrade is given by p = kw = — • The 

variation of these quantities along a meridional section through the deflection 
surface of the plate is shown in Fig. 131, together with similar curves based on a theory 
which will be discussed in Art. 61. 

At the origin we have kei x = — tt/4 and the deflection under the load becomes 

— - i <,80) 

For the reaction of the subgrade at the same point we obtain 

?w = £ 5 (181) 



2 



81 

If we take an infinitely large plate with the conditions of rigidity and loading assumed 
on page 264, the deflection under the load becomes 

uw = ^ = £f 2 = *IA< = (3.14)(5)(102 • 10-) = 0.016 in. 

as compared with the value of 0.02 in. obtained for a finite circular plate with the 

radius a = 23. 

The distribution of the bending moments due to the concentrated load is shown in 



268 



THEORY OF PLATES AND SHELLS 



"-w— ^(Elast. solid) 




^ 6 -»r/l =x 

-*r/l=x 




-0.125 (Hertz) 



I = 0.806 1 



(b) 



r/lo= x 
r/l=x 



0.05 



P 



0.10 



0.15 




v=o.z 



x = x 



(c) 
Fig. 131 

Fig. 131c. It is seen that the radial moments become negative at some distance from 
the load, their numerically largest value being about — 0.02P. The positive moments 
are infinitely large at the origin, but at a small 1 distance from the point of application 
of the load they can be easily calculated by taking the function kei x in the form (i). 
Upon applying formulas (52) and (53) to expression (179), we arrive at the results 

1 As compared with the characteristic length I = \^D/k. 



PLATES ON ELASTIC FOUNDATION 269 

M r -£[a + ,)(iogf - T )-I(i-,)1 

r / \ i i (182) 

Mt =£ (1 +^)(log^-7j + ^(l -ol 

A comparison of the foregoing expressions with Eqs. (90) and (91) shows that the 
stress condition in a plate in the vicinity of the load in Hertz's case is identical with 
that of a simply supported circular plate with a radius a = 2le~t = 1.1 23Z, except for 

p 
a moment M' r = M \ = (1 — v), which is superimposed on the moments of the 

circular plate. 

Let us consider now the case in which the load P is distributed over the area of a 
circle with a radius c, small in comparison with /. The bending moments at the center 
of a circular plate carrying such a load are 



M r = M t = — 
4?r 



H 



(1 + i01og-"+ 1 ( m ) 



This results from Eq. (83), if we neglect there the term c 2 /a 2 against unity. By 
substituting a = 2le~v into Eq. (m) and adding the moment -P/8tt(1 - v), we obtain 
at the center of the loaded circle of the infinitely large plate the moments 

(1 + v)P / 21 l\ , , 

M max = ^^ (log I + 0.616^ (183) 

Stresses resulting from Eq. (183) must be corrected by means of the thick-plate theory 
in the case of a highly concentrated load. Such a corrected stress formula is given on 

page 275. 

In the case of a load uniformly distributed over the area of a small rectangle, we may 
proceed as described in Art. 37. The equivalent of a square area, in particular, is a 
circle with the radius c = 0.57u, u being the length of the side of the square (see page 
162). Substituting this into Eq. (183) we obtain 



Hog ~ + 1.177 \ 



M max = l -~- P [ log - + 1.177 1 (o) 



The effect of any group of concentrated loads on the deflections of the infinitely 
large plate can be calculated by summing up the deflections produced by each load 
separately. 

59. Rectangular and Continuous Plates on Elastic Foundation. An 

example of a plate resting on elastic subgrade and supported at the same 
time along a rectangular boundary is shown in Fig. 132, which represents 
a beam of a rectangular tubular cross section pressed into an elastic 
foundation by the loads P. The bottom plate of the beam, loaded by 
the elastic reactions of the foundation, is supported by the vertical sides 
of the tube and by the transverse diaphragms indicated in the figure by 



270 



THEORY OP PLATES AND SHELLS 



dashed lines. It is assumed again that the intensity of the reaction p at 
any point of the bottom plate is proportional to the deflection w at that 
point, so that p = kw, k being the modulus of the foundation. 

In accordance with this assumption, the differential equation for the 
deflection, written in rectangular coordinates, becomes 




d 4 w 
dx 4 



+ 2 



d 4 W 



+ 



d 4 w 



dx 2 dy* ' dy 4 



= Q_ _ 
D 



kw 
~D 



(a) 



where g, as before, is the intensity of the lateral 
load. 

Let us begin with the case shown in Fig. 132. 
If wo denotes the deflection of the edges of the 
bottom plate, and w the deflection of this plate 
with respect to the plane of its boundary, the 
intensity of the reaction of the foundation at 
any point is /c(w — w), and Eq. (a) becomes 



. |./W 



k 

AAW = jz (Wq — V)) 



(b) 



T 



Fig. 132 



Taking the coordinate axes as shown in the 
figure and assuming that the edges of the plate 

parallel to the y axis are simply supported and the other two edges are 

clamped, the boundary conditions are 

(*>— .~ = ° (0) x= 



= o 



= 0,2= 



(w)*-±6/2 = (-T-) =0 

VZ/A=±!>/2 

The deflection w can be taken in the form of a series: 



(c) 

(d) 



w = 






00 

■I 

r»= 1,3,5, . 



sin 



rrnrx 
a 



/ mV 4 k\ 
U 4 + D) 



+ 



m 



00 



Sill 



rrnrx 



a 



(e) 



m= 1,3,5, ... 



The first series on the right-hand side is a particular solution of Eq. (6) 
representing the deflection of a simply supported strip resting on an 
elastic foundation. The second series is the solution of the homogene- 
ous equation 



k 
AAw + -= w = 



(/) 



Hence the functions Y m have to satisfy the ordinary differential equation 



yiv 

1 m 



mV 2 „ 



(mV k \ 



Y m = 



(g) 



PLATES ON ELASTIC FOUNDATION 271 

Using notations 

m7r k -, \4 th\ 

V = "" D X {k) 

2& = Vn* + X 4 + ^ 2 7 ^ = VmI + X 4 - M i (0 

and taking the solution of Eq. (g) in the form e r2/ , we obtain for r the 
following four roots: 

+ iy — £ + £y — iy — /3 — iy 

The corresponding four independent particular solutions of Eq. (g) are 

e e m y cos Jm y e -fi m v C os T m ?/ efi my sin 7m2/ e~" /Jm2/ sin y m y (j) 

which can be taken also in the following form: 

cosh p m y cos y m y sinh /3 m y cos 7m?/ /^n 

cosh /3 m y sin 7 m ?/ sinh fl m y sin 7 m y 

From symmetry it can be concluded that Y m in our case is an even func- 
tion of y. Hence, by using integrals (/c), we obtain 

Y m = A m cosh p m y cos 7m?/ + B m sinh ]8 w i/ sin y m y 

and the deflection of the plate is 



w = 

m = l,3,5, 



00 j- 

S. m7rx 4kwo 1 
sin ^ — t ~v 



+ Am cosh j8 TO 2/ cos y m y + 5 m sinh /3 m i/ sin y m y 



0) 



This expression satisfies the boundary conditions (c). To satisfy con- 
ditions (d) we must choose the constants A m and B m so as to satisfy the 
equations 

^-° —7—1 C + A. CO S h ^ COS ^ 

JL/7T 

m 



/mV 4 . AA 



+ B m smh- T sm^- = (w) 



(A w j8 w + 5 m 7 m ) sinh -|- cos -~ 



— (4«t« — B m p m ) cosh -|- sin -g- = 

Substituting these values of A m and B m in expression (I), we obtain the 
required deflection of the plate. 

The problem of the plate with all four edges simply supported can be 
solved by using Eq. (a). Taking the coordinate axes as shown in Fig. 59 



272 THEORY OF PLATES AND SHELLS 

(page 105) and using the Navier solution, the deflection of the plate is 

oo oo 

w = 2/ L, Amn sm ir sm x (n) 

m = l n=l 

In similar manner let the series 



\ \ . mwx . mry 

q = 2/ 2/ amn Sm ^~ Sm V (o > 

m = l n = l 

represent the distribution of the given load, and the series 

* = kw = X 2 kAmn sin ^ sin t <p> 

represent the reaction of the subgrade. Substituting the series (n) in 
the left-hand side and the series (o) and (p) in the right-hand side of 
Eq. (a), we obtain 

as an example, let us consider the bending of the plate by a force P 
concentrated at some point (f^). In such a case 

4P . mir£ . rnrr) 
dmn = — r sm — ~ sm ~-r-' O) 

by Eq. (6) on page 111. By substitution of expressions (q) and (r) into 
Eq. (n) we finally obtain 

. ?7l7r£ . UT7I 

in sm — =-* 

Ob 7 j ?, ._/««. ««V Sm IT Sm b « 



w = ^\ \ « *L_ sin !^ sin ?^ 



Having the deflection of the plate produced by a concentrated force, 
the deflection produced by any kind of lateral loading is obtained by the 
method of superposition. Take, as an example, the case of a uniformly 
distributed load of the intensity q. Substituting q d£ dr) for P in expres- 
sion (s) and integrating between the limits and a and between and b, 
we obtain 



%t t 



. rrnrx . mry 
sm sm -~ 

w = ^? > > = « b 



m= 1,3,5,... n = l,3,5,... 



Ms +#'+»] 



PLATES ON ELASTIC FOUNDATION 273 

When k is equal to zero, this deflection reduces to that given in Navier 
solution (131) for the deflection of a uniformly loaded plate. 1 

Let us consider now the case represented in Fig. 133. A large p)ate 
which rests on an elastic foundation is loaded at equidistant points along 
the x axis by forces P.* We shall take the coordinate axes as shown in 




Fig. 133 

the figure and use Eq. (/), since there is no distributed lateral load. Let 
us consider a solution of this equation in the form of the series 

00 

w = wo + )_j m C0S ~^T 

m = 2,4,6, ... 

in which the first term 



P\ 

Wq = 



2 V2 ak 



,-W^/cos^ + sin^ 



represents the deflection of an infinitely long strip of unit width parallel 
to the y axis loaded at y = by a load P/a [see Eq. (283), page 471]. 
The other terms of the series satisfy the requirement of symmetry that 
the tangent to the deflection surface in the x direction shall have a zero 
slope at the loaded points and at the points midway between the loads. 
We take for functions Y m those of the particular integrals (J) which 
vanish for infinite values of y. Hence, 

Y m = A m er* mV cos y m y + B m e-^ v sin y m y 

To satisfy the symmetry condition (dw/dy) y==i o = we must take in this 
expression 

1 The case of a rectangular plate with prescribed deflections and moments on two 
opposite edges and various boundary conditions on two others was discussed by H. J. 
Fletcher and C. J. Thorne, J. Appl. Mechanics, vol. 19, p. 361, 1952. Many graphs 
are given in that paper. 

* This problem has been discussed by H. M. Westergaard; see Ingenifren, vol. 32, 
p. 513, 1923. Practical applications of the solution of this problem in concrete road 
design are discussed by H. M. Westergaard in the journal Public Roads, vol. 7, p. 25, 
1926; vol. 10, p. 65, 1929; and vol. 14, p. 185, 1933. 



274 THEORY OF PLATES AND SHELLS 

Hence, by introducing the new constants A' m = A m /-f m , we represent the 
deflections (u) in the following form: 

00 

w = Wo+ 2j A '™ cos ^ e-^y ( 7 m cos y m y + m sin y m y) (v) 

m = 2,4,6, ... 

In order to express the constants A' m in terms of the magnitude of loads 
P, we consider the shearing force Q y acting along the normal section of 
the plate through the x axis. From symmetry we conclude that this 
force vanishes at all points except the points of application of the loads 
P, at which points the shearing forces must give resultants equal to 
— JP/2. It was shown in the discussion of a similar distribution of 
shearing forces in Art. 54 (see page 248) that the shear forces can be 
represented by the series 

00 

m = 2,4,6, ... 

The shearing force, as calculated from expression (v), is 



rrnrx 



Qv dy \*~* + 




dx 1 ay / y =o 

00 

A'J3 m y m (fil + tD cos ^- r 

a 

m = 2,4,6,... 



- - Ta - 2D 



Comparing these two expressions for the shearing force, we find 



AL = 



p(_!)«./2 



2aD(3 m y m (Pl + 7 i) 
or, by using notations (i), 

A , _ P(-l)«/t 

m aD\ VK + £ 
Substituting this in expression (v), we finally obtain 

, PX 2 V (-1)*/* rmrx . . 

W = W » + -aT 2/ VPTS C ° S ~^~ (Tm C ° S 7mV 

m<= 2,4,0, ... *" 

+ &» sin y m y) (w) 

The maximum deflection is evidently under the loads P and is obtained 
by substituting x = a/2, 2/ = in expression (w), which gives 

PX . PX 2 






2V20A; 

m = 2,4,'6, . . 



PLATES ON ELASTIC FOUNDATION 275 

The deflection in the particular case of one isolated load P acting on 
an infinitely large plate can also be obtained by setting a = °o in formula 
(184). In such a case the first term in the formula vanishes, and by using 
notations {%) we obtain 

PX 2 V 2tt / Vx 4 + m^--"^ 

^"^V^k L, *V * 4 + ti 

w = 2,4,6, ... 



/N 



PX 2 /"- /VX 4 + M 4 -M_ 2 d 



2V2irkJo > X4 + ^ 4 



Using the substitution 

we find 

WW* = 



M 2 



X 2 2m Vm 2 +.. 1 

PX 2 /*" 1 dw PX 2 



/•"J. 

Jo V2l 



2V2xfcjo V5"f+«' 8fc (185) 

in accordance with the result (180). With this magnitude of the deflec- 
tion, the maximum pressure on the elastic foundation is 



r ^ _ t _ Px2 _ L IE 



(186) 



The maximum tensile stress is at the bottom of the plate under the point 
of application of the load. The theory developed above gives an infinite 
value for the bending moment at this point, and recourse should be had 
to the theory of thick plates (see Art. 26). In the above-mentioned 
investigation by Westergaard the following formula for calculating maxi- 
mum tensile stress at the bottom of the plate is established by using the 
thick-plate theory: 

P Eh z 

(tr r ) max = 0.275(1 4- v) ^ logio w 0*0 

Here ft denotes the thickness of the plate, and 



b = VI .6c 2 + ft 2 - 0.675/1 when c < 1.724ft 
= c when c> 1.724ft 

where c is the radius of the circulai area over which the load P is assumed 
to be uniformly distributed. For c = the case of the concentrated 
force is obtained. 

In the case of a square loaded area u by u, we have to replace c by 
0.57u (see page 162). 

The case of equidistant loads P applied along the edge of a semi-infinite 
plate, as shown in Fig. 134, can also be treated in a similar way. The 
final formula for the maximum tensile stress at the bottom of the plate 



276 



THEORY OF PLATES AND SHELLS 



under the load when the distance a is large is 

«L« = 0.529(1 + 0.54.) ^ Tlogxo (0) - 0.71 j 



(y) 



where b is calculated as in the previous case, and c is the radius of the 
semicircular area over which the load P is assumed to be uniformly 
distributed. Formulas (x) and (y) have proved very useful in the design 
of concrete roads, in which case the 
circle of radius c represents the area 
of contact of the wheel tire with the 
road surface. 1 



B2 



v. 



a 



flr* 



II 




>|C\J 



>|CM 



WA 



E2 



\u 



E& 



Wa 



v, 



>, 



& 



60. Plate Carrying Rows of Equidistant 
Columns. As a last example, let us con- 
sider an infinite plate or mat resting on 
elastic subgrade and carrying equidistant 
and equal loads P, each load being distrib- 
uted uniformly over the area u by v of a 
rectangle, as shown in Fig. 135. The 

yy_^-^ 

Fig. 134 

bending of such a " reversed flat slab" may be treated by means of the previously dis- 
cussed Westergaard's solution, using simple series. 2 Much simpler, however, and, 
except for the case of a highly concentrated load, also adequate is the solution in 
double series, making use of Navier's method. 

Conditions of symmetry compel us to represent the lateral load due to the columns 
in form of a cosine series: 



Fig. 135 



Q = 



22- 

m=0 n=0 



2mirx 2niry 

mn COS COS — : 



(fl) 



The intensity of the given load is equal to P/uv within the shadowed rectangles in 

Fig. 135 and is zero elsewhere. Thus, proceeding in the usual manner, i.e., multiplying 

™ / n i. 2mirx 2mcy 

H.q. (a) by cos cos — — ax ay and integrating between the limits —a/2, +a/2 

a o 

1 The problem of stress distribution near the load applied at a corner of a large plate 
has not yet been solved with the same reliability as the problems discussed above. 
Several empiric and semiempiric stress formulas regarding that case may be found in 
"Concrete Pavement Design," p. 79, Portland Cement Association, Chicago, 1951. 
Noteworthy experimental results concerning this problem were obtained by M. Dantu, 
Ann. ponts et chauss&es, vol. 122, p. 337, 1952. See also L. D. Black, Trans. Eng. 
Inst. Canada, vol. 2, p. 129, 1958, and D. E. Nevel, ibid., p. 132. 

2 See W. Muller, Ingr.-Arch., vol. 20, p. 278, 1952, and Osterr. Ingr.-Arch., vol. 6, 
jp ? 404, 1952. 



PLATES ON ELASTIC FOUNDATION 277 

for x; —6/2, +6/2 for y, we have 

4P€ mn . rrnru . mrv x 

a wn = -r sin sin — (o) 

ir 2 mnuv a o 

where c mn = 1 for m ^ 0, n ^ 

e mB = £ for ra = 0, n ?£ or m 5^ 0, n = 

% = i for ra = n = 
In the particular case of m = or n = the coefficient itself is readiiy obtained as a 
limit value of the expression (6). 

Now, in accordance with Eq. (a) we take for deflections the series 



00 00 

■11 



2mirx 2mry 

W = / / A mn COS COS — — (C) 



m 



= n = 



and the relation between the coefficients a mn and A mn is easily established by the same 
reasoning as before (see page 272). Thus, using the notation 

2m<ir 2ut 2 2 2 , ,. 

Ot m = Pn = -J— Tmn = <*m + Pn «*> 

a 

we obtain 

A — Q ^ w /' \ 

i>7mn + * 

Substituting this in the series (c) and observing Eq. (6) we have the final result 1 

rrnru . n-nv 
* JL emu sin sin — cos a m x cos j3 n i/ 

W = ^VV ° ^4 . ,, (/) 



TT 2 UV 

m = n 



2, 2/ tnn(D 7 L + *) 



The bending moments of the plate are now obtained by the usual differentiation, and 
the distribution of the pressure between the plate and the subgrade is found by multi- 
plication of expression (/) by the modulus &. 

The particular case k = corresponds to a uniformly distributed reaction of the 
subgrade, i.e., to the case of a " reversed flat slab" uniformly loaded with q = P/ab. 
It is seen from Eq. (/) that the introduction of the modulus tends to reduce the deflec- 
tions and also the bending moments of the plate. 

The case of a rectangular plate of finite dimensions resting on an elastic foundation 
and submitted to the action of a concentrated load has been discussed by H. Happel. 2 
The Ritz method (see page 344) has been used to determine the deflections of this 
plate, and it was shown in the particular example of a centrally loaded square plate 
that the series representing the deflection converges rapidly and that the deflection 
can be calculated with sufficient accuracy by taking only the first few terms of the 
series. 3 

1 Due to V. Lewe, Bauingenieur, vol. 3, p. 453, 1923. 

2 Math. Z., vol. 6, p. 203, 1920. See also F. Halbritter, Bautechnik, vol. 26, p. 181, 

1949. 

3 The problem of a square plate on an elastic foundation has also been investigated 
experimentally; see the paper by J. Vint and W. N. Elgood, Phil. Mag., ser. 7, vol. 19, 
p. 1, 1935; and that by G. Murphy, Iowa State Coll. Eng. Expt. Sta. Bull. 135, 1937. 



278 



THEORY OF PLATES AND SHELLS 



61. Bending of Plates Resting on a Semi-infinite Elastic Solid. So far, the settling 
of the subgrade at some point of its surface has been assumed as proportional to the 
pressure between the plate and the subgrade at the same point, and consequently as 
independent of the pressure elsewhere. This is correct in the case of a floating plate, 
considered by Hertz (see page 260), but in the case of a coherent subgrade such a 
hypothesis approximates but crudely the actual behavior of the subgrade; a better 
approximation can sometimes be obtained on the basis of the following assumptions: 

1. The foundation has the properties of a semi-infinite elastic body. 

2. The plate rests on the subgrade without friction. 

3. A perfect contact between the plate and foundation also exists in the case of a 
negative mutual pressure. 

This last supposition appears arbitrary; however, a negative pressure between plate 
and subgrade actually is compensated, more or less, by the weight of the plate. 

The elastic properties of the elastic foundation may be characterized, if isotropy is 
assumed, by a Young modulus E and a Poisson ratio v . The approximate numerical 
values 1 of these constants, depending on the nature of the subgrade and based on 
results of dynamical tests, are given in Table 63, together with the value of the 
constant 

&o = — ^ (a) 



used in the following. 



2(1 - rj) 



Table 63. Values of Elastic Constants Depending on 
Nature of Foundation 



Subgrade 


E 0> psi 


Vo 


ko, psi 


Clay 


11,000 

13,000 

14,000-18,500 

40,000 

38,000 

165,000-190,000 

1,600,000 


0.17 

0.42 
0.33-0.23 

0.31 

0.44 
0.32-0.38 

0.26 


5,700 


Loess and clay 


7,900 


Medium sand 


7,900-9,800 


Sand and gravel 


22,000 
23,500 
92,000-110,000 
860,000 


Liassic plastic clay 

Lime (air-slaked) 

Sandstone 



We restrict the further consideration to the case of an infinitely large plate in a state 
of axial symmetry. Using polar coordinates r, 0, we can write the plate equation as 

DAAw(r) = q(r) — p(r) (b) 

where q(r) denotes the given surface loading and j>(r) the reaction of the subgrade. 

Let i£ (r,p,^>) be the deflection at the point (r,0) of the subgrade surface due to a 
normal unit load applied on this surface (p,^>). The form of the "influence function'' 
Ko depends merely upon the nature of the foundation. Making use of some properties 
of the Bessel functions, it can be shown 2 that Eq. (6) is satisfied by the expression 



w(r) 



-/.• 



Q(a)K(a)Jo(ar)a da 
1 + Da*K(a) 



(C) 



1 Due to E. Schultze and H. Muhs, " Bodenuntersuchungen fur Ingenieurbauten," 
Berlin, 1950. See also Veroffentl. Degebo, Heft 4, p. 37, 1936. 

2 The solution of the problem in this general form is due to D. L. Holl, Proc. Fifth 
Intern. Congr. Appl. Mech., Cambridge, Mass., 1938. 



PLATES ON ELASTIC FOUNDATION 279 

n Eq. (c) Jo denotes the Bessel function of zero order; the term depending on the 
mture of the subgrade is 

K( a ) = f°° 2wsK l) (s)Jo(as)ds (d) 

n which the form of Ko is defined by 

Ko(s) = X [(r 2 + p 2 - 2r P cos <p)h] 
? being the distance between points (r,0) and (p,v?). Finally 

Q(a) = f" q(p)Jo(« P )p dp (e) 

Is the term depending on the intensity q(p) of the symmetrical loading at r = p. 

In the particular case of a load P uniformly distributed along the periphery of a 
-irele with a radius c, we have 

Q(a) = —Mac) (/) 

2tt 

In the case of the load P distributed uniformly over the area of the same circle, Eq. 
(e) yields 

P 

Q(a) = Ji(ac) KQ) 

irCa 

where the Bessel function is of the order one. Finally, where a load is concentrated 
at the origin (p =0), we obtain from Eq. (/) 

Q(a) = ^ (h) 

As for the distribution of the reactive pressure, the respective function p(r) is 
obtained from Eq. (b), the term 

q (r) = J Q(<x)J (ar)ada (0 

being previously expressed through its Fourier-Bessel transform (e). Thus, we obtain 

[« Q(a).h(ar)a da 

Now let us consider two particular cases with respect to the physical nature of the 
subgrade. For & floating plate (Art. 57) the influence function K (s) is zero everywhere 
except at s = 0, where the unit force is applied. With regard to Eq. (d) the quan- 
tity K (a) then must be a constant. In order to get from Eq. (c) the expression. 
w (r) = p(r)/k, this in accordance with the definition of the modulus, we have to 
assume Ko(a) = 1/k. Using the previous notation Z 4 = D/k (page 260), we obtain 
from Eq. (c) the expression 

/N lf°° Q(a)Jo(ar)ada . 

w(r)=-J o —Yy^r- (fc) 

which actually satisfies the differential equation (178) of the floating plate. 

In the case of an isotropic semi-infinite medium we have, by a result due to Bous- 



280 THEORY OF PLATES AND SHELLS 

sinesq, 1 K (s) *= (1 » 4)/ttE s and, by Eq. (d), K(a) = 2(1 - vl)/Eoa, or 

K(a) = -i- 
(k Q a) 

where k is the elastic constant defined by Eq. (a). Writing for brevity, 

ko Eq 1 

D = 22>(1 - vl) = $ (/ 

we finally obtain the solution (c) in the more special form 2 

w ^ = r / — r~; — i75 — (m: 

ft Jo 1 + a 3 Z; 

In the particular case of a load concentrated at the origin, expression (m) in con- 
nection with (h) yields 



Jo l+ x 3 



Pit ' "°«^ Jdx 



w = ^Jo "T+1?- (187) 

where X is written for «i . Therefore, the deflection under the load is 

Pll /"" d\ PllVl Pl\ 

against the result 0.125Pl*/D of Hertz. The distribution of the pressure is readily 
obtained from the general expression (j). We have at any point 



p_ [ J °(r) xdx 

p arfj Jo i + x 3 



(189) 



and especially under the load 



P f" XdX P \/z P 

»~-*nJo rr^ = ^r = - 192 ]i (190) 

in comparison with the value of 0.125P/Z 2 obtained by Hertz. If we assume equal 
values of w max in both cases, formula (190) yields a value for p max which is 2.37 times 
as large as the value from Hertz's formula (181). In such a case the relation 
I = 1.241Z must hold, and curves of the respective deflections as calculated from 
Eqs. (179) and (187) are shown in Fig. I31a. Figure 1316 shows in like manner the 
variation of the pressure; this time, in order to obtain equal values for p max in both 
cases, it must be assumed that I = 0.806Z . 

It can be shown, finally, that the magnitude of bending moments in the vicinity of 

^ee, for example, S. Timoshenko and J. N. Goodier, " Theory of Elasticity," 
2d ed., p. 365, New York, 1951. 

2 For this result see also S. Woinowsky-Krieger, Ingr.-Arch., vol. 3, p. 250, 1932, and 
vol. 17, p. 142, 1949; K. Marguerre, Z. angew. Math. Mech., vol. 17, p. 229, 1937; 
A. H. A. Hogg, Phil. Mag., vol. 25, p. 576, 1938. 



PLATES ON ELASTIC FOUNDATION 281 

the concentrated load is the same for foundations of both kinds if expressed in terms 
of the dimensionless argument x = r/l and x — r/U, respectively. We conclude from 
this fact that expressions for bending moments, such as given by Eq. (183), can also be 
used for a plate resting on an isotropic elastic medium if we replace I by Z . Proceeding 
in this manner with the stress formula (x) of Westergaard (page 275), we arrive at the 
formula 

P 



<r max = 0.366(1 + v) hi 



1Ogl0 (S)-°- 266 ] W 



in which k is given by Eq. (a), and b denotes the same quantity as on page 275. 

The problem of the bending of a finite circular plate leads to an infinite set of linear 
equations for the coefficients of the series, which has to represent the deflections of 
such a plate. 1 

The use of the method of finite differences should also be considered in handling the 
problem of finite circular plates. 2 

The bending of an infinite plate supported by an elastic layer, which rests in its turn 
on a perfectly rigid base, 3 and the problem of a semi-infinite pavement slab 4 have also 
been discussed. 

Stresses due to a highly concentrated surface load should be corrected in accordance 
with the general theory of thick plates. However, a special theory of thick plates 
supported elastically has also been established. 5 

x See H. Borowicka, Ingr.-Arch., vol. 10, p. 113, 1939; A. G. Ishkova, Doklady 
Akad. Nauk S.S.S.R., vol. 56, p. 129, 1947; G. Pickett and F. J. McCormick, Proc. 
First U.S. Natl. Congr. Appl. Mech., p. 331, Chicago, 1951. The effect of raising the 
outer portion of the plate submitted to a central load was discussed by H. Jung, 
Ingr.-Arch., vol. 20, p. 8, 1952. For bending of rectangular plates see M. I. Gor- 
bounov-Posadov, Priklad. Mat. Mekhan., vol. 4, p. 68, 1940. 

2 A. Habel, Bauingenieur, vol. 18, p. 188, 1937; for application to rectangular plates 
see G. Pickett, W. C. Janes, M. E. Raville, and F. J. McCormick, Kansas State Coll. 
Eng. Expt. Sta. Bull. 65, 1951. 

3 A. H. A. Hogg, Phil. Mag., vol. 35, p. 265, 1944. 

4 G. Pickett and S. Badaruddin, Proc. Ninth Intern. Congr. Appl. Mech., vol. 6, 
p. 396, Brussels, 1957. 

5 The first discussion of the statical and dynamical behavior of such plates is due to 
K. Marguerre, Ingr.-Arch., vol. 4, p. 332, 1933; see also I. Szabo, Ingr.-Arch., vol. 19, 
pp. 128, 342, 1951; Z. angew. Math. Mech., vol. 32, p. 145, 1952. For application of 
E. Reissner's theory see P. M. Naghdi and J. C. Rowley, Proc. First Midwest Conf. 
Solid Mech. (Univ. Illinois), 1953, p. 119, and D. Frederick, J. Appl. Mechanics, 
vol. 23, p. 195, 1956. 



CHAPTER 9 



PLATES OF VARIOUS SHAPES 



62. Equations of Bending of Plates in Polar Coordinates. In the 

discussion of symmetrical bending of circular plates polar coordinates 
were used (Chap. 3). The same coordinates can also be used to advan- 
tage in the general case of bending of circular plates. 

If the r and coordinates are taken, as shown in Fig. 136a, the relation 
between the polar and cartesian coordinates is given by the equations 



x 1 + ?/ 2 



= arctan - 
x 



(a) 



from which it follows that 



or 
dx 


= 


X 

r 


d0 
dx 


= 





cos 



y 



sin 



dr 

dy 
d0 
dy 



y 

r 
x 



sin 
cos 



(P) 



Using these expressions, we obtain the slope of the deflection surface of a 

plate in the x direction as 

dw __ dw dr dw 80 

dx dr dx d0 dx 

dw 1 dw . f . 

= — cos — sin (c) 

dr r d0 w 

A similar expression can be written 
for the slope in the y direction. 
To obtain the expression for curva- 
ture in polar coordinates the second derivatives are required. Repeating 
twice the operation indicated in expression (c), we find 




Fig. 136 



d 2 w 
dx* 



-(- 

\dr 



cos 6 sin 6 — ) { -r— cos 6 

r dd/\dr 

d 2 w , n o d^w sin B cos 6 , dw sin 2 9 

dr 2 36 dr r dr r 



Idw . A 
r dd ) 



. n dw sin 6 cos 6 , d 2 w sin 2 6 ,,. 

+ 2 T7T Z5 1" ^T — 3— W 



dd 



de 2 



282 



PLATES GIT VARIOUS SHAPES 283 

'n a similar manner we obtain 

\ 2 w d 2 w . n A , n d 2 w sin cos . dw cos 2 

^ = ^ SmH ^ r + dF~r- 

dw sin cos dhv cos 2 , v 
- ^ ^ — r 2 + ^2 f 2 W 

d 2 w d 2 w . „ „ , 6 2 tf cos 20 dw cos 20 

l JT" = VT sin # C0S " + 3 a/, "^ ~2 

to; dy dr 2 dr 30 r 80 r' 

dw sin cos d 2 w sin cos 



dr r dd 2 r 2 

With this transformation of coordinates we obtain 



(/) 



d 2 W , d 2 W d 2 W , 1 dw , 1 d 2 W ,* 

Repeating this operation twice, the differential equation (103) for the 
leflection surface of a laterally loaded plate transforms in polar coordi- 
lates to the following form: 

/d 2 .1 d ,1 d 2 \/d 2 w ldw 1 d 2 w\_q_ nm 

AAw = \W 2 + 7Yr + F 2 dT 2 )\dF^~rlF + F 2 de 2 ~)~ D { ™ l) 

When the load is symmetrically distributed with respect to the center of 
the plate, the deflection w is independent of 0, and Eq. (191) coincides 
with Eq. (58) (see page 54), which was obtained in the case of sym- 
metrically loaded circular plates. 

Let us consider an element cut out of the plate by two adjacent axial 
planes forming an angle dd and by two cylindrical surfaces of radii r and 
, _j_ dr, respectively (Fig. 136b). We denote the bending and twisting 
moments acting on the element per unit length by M r , M h and M rf and 
bake their positive directions as shown in the figure. To express these 
moments by the deflection w of the plate we assume that the x axis coin- 
sides with the radius r. The moments M r , Mi, and M rt then have the 
same values as the moments M x , M y , and M v at the same point, and by 
substituting = in expressions (d), (e), and (/), we obtain 

^ (&™ . 3 2 wA n \d 2 w (I dw 1 d 2 w\] 

^ /d 2 w . d 2 w\ ~ /l dw . 1 d 2 w , d 2 w\ n no . 

M - = - D {w + '^)>.r- D {^ + ^^ + '^) (192) 

v ^ / d 2 W \ ,„ , n /l d 2 W 1 dw\ 

In a similar manner, from formulas (108), we obtain the expressions for 



284 THEORY OF PLATES AND SHELLS 

the shearing forces 1 

Q r - - D jL (A») and Q t = - D ^ (108) 

where Aid is given by expression (0). 

In the ease of a clamped edge the boundary conditions of a circular 
plate of radius a are 

(«0« = o (^) =0 (*) 



In the case of a simply supported edge 

Wr=a = (M r )r=a = (i) 

In the case of a free edge (see page 87) 

(M r ) r==o = 7-Yq,-^ =0 0")' 



The general solution of Eq. (191) can be taken, as before, in the form 
of a sum 

w = wo + wx (k) 

in which Wo is a particular solution of Eq. (191) and Wi is the solution of 
the homogeneous equation 

(*.+l*. + L*.\ (**H + 1^ + 1^=0 (194) 
\dr 2 T rdr^ r 2 86*) \dr 2 T r dr T r 2 30 2 / *■ ' 

This latter solution we take in the form of the following series: 2 

co go 

W\ = Ro + J R m cos md + y ^ sin md (195) 

in which Ro, R ly . . . , i2f , R' 2 , . . . are functions of the radial distance 
r only. Substituting this series in Eq. (194), we obtain for each of these 
functions an ordinary differential equation of the following kind : 

/j*l + I §l _ !Hl\ ( d2R ™ i I ^s _ ™> 2 R m \ 
\dr 2 r dr r 2 J \ dr 2 r dr r 2 J 

The general solution of this equation f or m > 1 is 

R m = A m r™ + B m r-> + Ci"** + D m r-> + 2 (0 

1 The direction of Q r in Fig. 1366 is opposite to that used in Fig. 28. This explains 
the minus sign in Eq. (193). 

2 This solution was given by A. Clebsch in his "Theorie der Elasticitat fester 
Korper," 1862. 



PLATES OF VARIOUS SHAPES 



285 



(m) 



For ra = and m — 1 the solutions are 

R - A + 5 r 2 + Co log r + ZV 2 log r 
and #i = Air + ^r 3 + C x r~ l + IV log r 

Similar expressions can be written for the functions R' m . Substituting 
these expressions for the functions R m and R' m in series (195), we obtain 
the general solution of Eq. (194). The constants A m , B m , . . . , D m in 
each particular case must be determined so as to satisfy the boundary- 
conditions. The solution R 0) which is independent of the angle 0, repre- 
sents symmetrical bending of circular plates. Several particular cases 
of this kind have already been discussed in 
Chap. 3. 

63. Circular Plates under a Linearly 
Varying Load. If a circular plate is acted 
upon by a load distributed as shown in Fig. 
137, this load can always be divided into 
two parts: (1) a uniformly distributed load 
of intensity i(p 2 + Pi) and (2) a linearly 
varying load having zero intensity along the 
diameter CD of the plate and the intensities 
— p and +p at the ends A and B of the 
diameter AB. The case of uniform load 
has already been discussed in Chap. 3. We 
have to consider here only the nonuniform 
load represented in the figure by the two shaded triangles. 1 

The intensity of the load q at any point with coordinates r and 6 is 




Fig. 137 



pr cos 6 

q = £ 



(a) 



The particular solution of Eq. (191) can thus be taken in the following 

form: 

. vr 6 cos 6 
Wo = A t —— 



a 



This, after substitution in Eq. (191), gives 

1 



A = 



192D 



Hence 



(&) 



pr b cos 
Wo = 192aD 

As the solution of the homogeneous equation (194) we take only the term 
of series (195) that contains the function Ri and assume 

Wl = (Air + B x r s + Ci*- 1 + Dir log r) cos 6 (c) 

» This problem has been discussed by W. Flugge, Bauingenieur, vol. 10, p. 221, 1929. 



286 THEORY OP PLATES AND SHELLS 

Since it is advantageous to work with dimensionless quantities, we intro* 
duce, in place of r, the ratio 

r 
p = a 

With this new notation the deflection of the plate becomes 

w = Wo + Wl = j||^ (p 5 + Ap + Bp z + Cp- 1 + Dp log p) cos (d) 

where p varies from zero to unity. The constants A, B, ... in this 
expression must now be determined from the boundary conditions. 

Let us begin with the case of a simply supported plate (Fig. 137). In 
this case the deflection w and the bending moment M r at the boundary 
vanish, and we obtain 

(tiOp-i = (ikf r ) ps=sl = (e) 

At the center of the plate (p = 0) the deflection w and the moment M T 
must be finite. From this it follows at once that the constants C and D 
in expression (d) are equal to zero. The remaining two constants A and 
B will now be found from Eqs. (e), which give 

(w)p=i = ^ (1 + A + B) cos = 

(iWV)p-i = - gg [4(5 + v) + 2(3 + „)B] cos = 

Since these equations must be fulfilled for any value of 0, the factors 
before cos must vanish. This gives 

1 + A + B = 
4(5 + ?) + 2(3 + v)B = 
and we obtain 

6 + v 6 + v 

Substituting these values in expression (d), we obtain the deflection w 
of the plate in the following form : 

» = TmnrB [7 + " - (3 + m cos ' w 

For calculating the bending moments and the shearing forces we substi- 
tute expression (/) in Eqs. (192) and (193), from which 

M r = & (5 + „)p(l - p 2 ) cos B 

M t - 48( ^ + - } p[(5 + f)(1 + 3.) - (1 + 5v)(3 + v)p>] cos 



PLATES OF VARIOUS SHAPES 287 



(h) 



Qr = 24( .[^ p) [2(5 + v) - 9(3 + ,)p 2 ] cos e 

Ql " " 24(lT7) Pl2(5 + " } ~ 3(3 + " )p2] Shl * 
It is seen that (M r ) max occurs at p = 1/V3 and is equal to 

{ r)u " " 72 Va 

The maximum value of Af t occurs at 

P = V(5 + y)"(T+^0/ V3(l + 5?) (3 + v) 

and is equal to 

f , n _ pa 2 (5 + p)(l + 3y) 
Ww»W 72 3 + v 

The value of the intensity of the vertical reaction at the boundary is 1 

_ dMrt pa a 

- y= - Q ' + Tde = 4: cosd 

The moment of this reaction with respect to the diameter CD of the 
plate (Fig. 137) is 

A [* /2 pa , 7ra 3 p 

4 / — - cos 6 a 2 cos 6 ad = — r- 11 

Jo 4 4 

This moment balances the moment of the load distributed over the plate 
with respect to the same diameter. 

As a second example, let us consider the case of a circular plate with a 
free boundary. Such a condition is encountered in the case of a circular 
foundation slab supporting a chimney. As the result of wind pressure, 
a moment M will be transmitted to the 
slab (Fig. 138) . Assuming that the reac- "*] k 

tions corresponding to this moment are 
distributed following a linear law, as -p i>^ ^ 
shown in the figure, we obtain the same JL ~ 
kind of loading as in the previous case; 

and the general solution can be taken in the same form (d) as before. 
The boundary conditions at the outer boundary of the plate, which is 
free from forces, are 

(M r ) P ^ = o (F), =1 = (a - j^l)^ = (*) 

The inner portion of the plate of radius b is considered absolutely rigid. 
It is also assumed that the edge of the plate is clamped along the circle 

1 The reaction in the upward direction is taken as positive. 




288 THEORY OF PLATES AND SHELLS 

of radius b. Hence for p = b/a = the following boundary conditio 
must be satisfied: 

Substituting expression (d) in Eqs. (t) and (j), we obtain the followinj 
equations for the determination of the constants: 

4(5 + v) + 2(3 + x)B + 2(1 - v)C + (1 + v)D = 
4(17 + v) + 2(3 + v)B + 2(1 - v)C - (3 - v)D = 

4j8 4 + 2/3 2 B - 2|8- 2 C + D = 
From these equations 

_ ^ 4(2 +>)+.(!- v)/3 2 (3 + fl 4 ) 

* ~ '* (3 + r) + (1 - v)P 

n _ o 4(2 + y )/3 4 - (3 + r)/3 2 (3 + /3 4 ) D = 12 
C " ~~* (3 + v) + (1 - v)0 4 

Substituting these values in expression (d) and using Eqs. (192) and (193), 
we can obtain the values of the moments and of the shearing forces. The 
constant A does not appear in these equations. The corresponding term 

in expression (d) represents the rotation of 

the plate as a rigid body with respect to the 

t M T diameter perpendicular to the plane of Fig. 

(a) a ""^ 138. Provided the modulus of the founda- 
tion is known, the angle of rotation can be 

/*- calculated from the condition of equilibrium 

L. ~- f\ of the given moment M and the reactions of 

(b) the foundation. 
Using expression (d), the case of a simply 

<p $ supported circular plate loaded by a moment 
i=5^| M at the center (Fig. 139a) can be readily 



^ 





M * solved. In this case we have to omit the 

(c ' term containing p 6 , which represents the dis- 

tributed load. The constant C must be taken 
equal to zero to eliminate an infinitely large deflection at the center. 
Expression (d) thus reduces to 

w = (Ap + Bp* + Dp log p) cos (fc) 

The three constants A, B } and D will now be determined from the follow- 
ing boundary conditions: 

(uO,-i =■ (M r )^i = 
-a f+J (Mrf)p-i sin d0 + a 2 f+J (Q r ),-i cos d0 + M = 

The first two of these equations represent the conditions at a simply sup- 



PLATES OF VARIOUS SHAPES 



289 



ported edge ; the last states the condition of equilibrium of the forces and 
moments acting at the boundary of the plate and the external moment M . 
From Eqs. (I) we obtain 

1 + v Ma ^ Ma 



A = 



1 + v Ma 
3 + v SttD 



B = 



3 + v SttD 



D = - 



4irZ> 



Hence 



Ma 



w = 



8ttZ>(3 + v) 



p[(l + v)(l - p 2 ) + 2(3 + v) log p] cos (m) 



Because of the logarithmic term in the brackets, the slope of the deflec 
tion surface calculated from expression (m) becomes infinitely large. To 
eliminate this difficulty the central portion of radius b of the plate may 
be considered as absolutely rigid. 1 Assuming the plate to be clamped 
along this inner boundary, which rotates under the action of the moment 
M (Fig. 1396), we find 



Ma 



w = 



8dD[(3 + !>) + (!- »O0 4 ] 



{- [(l + + - ")W 



+ (1 + v)(l - /3 2 ) 2 p + 2[(3 + v) + (1 - v)p)p log p 

- /3 2 [(1 + v)/3 2 - (3 + v)]p~ 1 } cos 6 



(n) 



where |8 = 6/a. When j8 is equal to zero, Eq. (n) reduces to Eq. (m), 
previously obtained. By substituting expression («) in Eq. (192) the 
bending moments M r and M< can be calculated. 

The case in which the outer boundary of the plate is clamped (Fig. 
139c) can be discussed in a similar manner. This case is of practical 
interest in the design of elastic couplings of shafts. 2 The maximum 
radial stresses at the inner and at the outer boundaries and the angle of 
rotation <p of the central rigid portion for this case are 

h „ , N h „ M 



(<r r )r_b = a'-j-E<p 

(JL 



(cr r ) r==a = a i - E(p 

(X 



azEh* 

where the constants a, a h and a 2 have the values given in Table 64 

Table 64 



j8 = b/a 


a 


<*i 


«2 


0.5 
0.6 
0.7 
0.8 


14.17 
19.54 
36.25 
82.26 


7.10 
12.85 
25.65 
66.50 


12.40 

28.48 

77.90 

314.00 



1 Experiments with such plates were made by R. J. Roark, Univ. Wisconsin Bull. 
74, 1932. 

2 H. Reissner, Ingr.-Arch., vol. 1, p. 72, 1929. 



290 



THEORY OF PLATES AND SHELLS 



64. Circular Plates under a Concentrated Load. The case of a load 
applied at the center of the plate has already been discussed in Art. 19. 
Here we shall assume that the load P is applied at point A at distance b 
from the center of the plate (Fig. 140). x Dividing the plate into two 
parts by the cylindrical section of radius b as shown in the figure by the 
dashed line, we can apply solution (195) for each of these portions of the 
plate. If the angle is measured from the radius OA, only the terms 
containing cos md should be retained. Hence for the outer part of the 
plate we obtain 



w 



= Ro + 2, ^" 



cos md 



m— 1 



where 



Ro 



A + B r* + Co log r + D r* log r 
Air -\- Bir s + CV -1 + D x r log r 



(a) 



(&) 



R m = A m r™ + B m r~ + C m r m + 2 + D m r~» + 2 



Similar expressions can also be written for the functions R f , R[, R' m 
corresponding to the inner portion of the plate. Using the symbols A' m , 

B' m , . . . instead of A m , B m , . . . for the con- 
stants of the latter portion of the plate, from the 
condition that the deflection, the slope, and the 
moments must be finite at the center of the plate, 
we obtain 

CJ = D' = 
C[ = D[ = 




B' m = jy m = o 



\z Hence for each term of series (a) we have to 

Fig. 140 determine four constants for the outer portion of 

the plate and two for the inner portion. 

The six equations necessary for this determination can be obtained 

from the boundary conditions at the edge of the plate and from the 

continuity conditions along the circle of radius 6. If the outer edge of 

the plate is assumed to be clamped, the corresponding boundary con- 



1 This problem was solved by Clebsch, op. ciL See also A. Foppl, Sitzber. bayer. 
Akad. Wiss.y Jahrg., 1912, p. 155. The discussion of the same problem by using 
bipolar coordinates was given by E. Melan, Eisenbau, 1920, p. 190, and by W. Flugge, 
"Die strenge Berechnung von Kreisplatten unter Einzellasten, ,, Berlin, 1928. See 
also the paper by H. Schmidt, Ingr.-Arch., vol. 1, p. 147, 1930, and W. Muller, Ingr.- 
Arch., vol. 13, p. 355, 1943. 



PLATES OF VARIOUS SHAPES 



291 



ditions are 



<»>— o (f)„-» 



(«) 



Denoting the deflection of the inner portion of the plate by w x and 
observing that there are no external moments applied along the circle of 
radius b, we write the continuity conditions along that circle as 

dw __ dWi d 2 W d 2 Wi 



W = W\ 



dr 



dr 2 



5r 2 



f or r = b 



id) 



The last equation is obtained from a consideration of the shearing force 
Q r along the dividing circle. This force is continuous at all points of the 
circle except point A, where it has a discontinuity due to concentrated 
force P. Using for this force the representation in form of the series 1 



56+2 cos m$ ) (e) 

m= 1 

and for the shearing force the first of the expressions (193), we obtain 

s 6 + 1 cos me ) y) 



D — (Aw) r==& — D— (Awi) r=b 
dr or 



m = l 



From the six equations (c), (d), and (/), the six constants can be calcu- 
lated, and the functions R m and R' m can be represented in the following 
form: 



Rq = 

fll = 

K™ = 



8ttP> 

p 



8irD 



\ r , + >, log - f + (£i±jg^l^ 

( „ + „ log * + <» • ± r ;>'f - *■> • 



Pb 3 
16ttZ> 

Pfr 3 
16ttZ) 
P6" 



1 2(a 2 - b 2 )r _ (2a 2 - b 2 )r 3 
~~r + a 2 6 2 a 4 6 2 

2(a 2 - b 2 )r (a 2 - b 2 ) 2 r 3 _ 4r 
a 2 b 2 + a 4 6 4 6 2 g 



4r , al 
■gi log ~ r \ 

s] 



8m(m - 1)^2) la 2m _ 



(m — l)i> 2 — ma 2 + (m — l)r 
m(m — 1) b-r % 



m + 1 



a- 



P6" 



8m(ra - 1)ttP) }a 2 '" _ 



+ 



a- 



r m \ m + 1 /J 



(m - 1)6 2 - ma 2 + ^„,_ fi 



52. 



H- (m - 1) 



r tn+2 



a 



2 m 



1 



ra 



m + 



fc 2 _ i M 2m l) 

la 2 m + 1\6/ Jj 



A This series is analogous to the scries that was used in the case of continuous plateB 
(see p. 248). 



202 



THEORY OF PLATE* A.ND SHELLS 



Using these functions, we obtain the deflection under the load as 

b 2)2 



f \ P ^ 



a' 



(196) 



tzzzzz 



For 6 = this formula coincides with formula (92) for a centrally loaded 

plate. The case of the plate with simply sup- 
ported edge can be treated in a similar 
manner. 

The problem in which a circular ring plate 
is clamped along the inner edge (r = 6) and 
loaded by a concentrated force P at the outer 
boundary (Fig. 141) can also be solved by 

using series (a). In this case the boundary conditions for the clamped 

inner boundary are 




(w) r=h = 



et - ° 



to) 



For the outer boundary, which is loaded only in one point, the conditions 
are 



(V)r 



(M r ) r =a = 

00 



cos md 



m = 1 



■) 



(h) 

Calculations made for a particular 
case b/a = -f show 1 that the largest 
bending moment M r at the inner 
boundary is 



(Mr) 



r=M=0 — 



■ 4 - 45 £ 




IlLl « 5a 

"T 5xr 7tt|tt— > 
I 4 JLl \0 



s r 



.la 



Fig. 142 



The variation of the moment along 

the inner edge and also along a 

circle of radius r = 5a/6 is shown 

in Fig. 142. It can be seen that 

this moment diminishes rapidly as the angle 0, measured from the point 

of application of the load, increases. 

The general solution of the form (a) may be used to advantage in 
handling circular plates with a system of single loads distributed sym- 
metrically with respect to the center of the plate, 2 and also in the case of 

1 H. Reissner, loc. cit. 

2 By combining such reactive loads with a given uniform loading, we may solve the 
problem of a flat slab bounded by a circle; see K. Hajnal-Konyi, "Berechnung von 
kreisformig begrenzten Pilzdecken, ,, Berlin, 1929. 



* 2 )(1 - 


?) 














* ? + £ 2 


- 2x£ 


cos 


e) log 


x 2 + e - 
i + x*e - 


- 2.r£ 

- 2x£, 


cos 

COS 


e 
e 



PLATES OF VARIOUS SHAPES 293 

annular plates. For circular plates having no hole and carrying but one 
eccentric load, simpler solutions can be obtained by the method of com- 
plex variables, 1 or, when the plate is clamped, by the method of inversion. 2 
In this latter case the deflection surface of the plate is obtained in the form 

Pa2 In 

+ (*' + «" - 2xl cos 6) log tVs'V-% ^] < 197 > 

where x = r/a and £ = 6/a (Fig. 140). Expression (197) holds through- 
out the whole plate and yields f or x = £, = 0, that is, under the load, 
the value (196), previously obtained by the series method. 

65. Circular Plates Supported at Several Points along the Boundary. Considering 
the case of a load symmetrically distributed with respect to the center of the plate, we 
take the general expression for the deflection surface 
in the following form: 3 

w = wq + Wi (a) 

in which wq is the deflection of a plate simply sup- 
ported along the entire boundary, and w\ satisfies the 
homogeneous differential equation 

AAwi = (b) 

Denoting the concentrated reactions at the points of -p . .« 

support 1, 2, 3, . . . by N h JV 2 , . . . , Ni and using 

series (h) of the previous article for representation of concentrated forces, we have 
for each reaction Ni the expression 

^Q+^cosm*) (c) 

771= 1 

where 0; = — yi 

ji being the angle defining the position of the support i (Fig. 143). The intensity 
of the reactive forces at any point of the boundary is then given by the expression 

1 The simply supported plate was treated in that manner by E. Reissner, Math. 
Ann., vol. Ill, p. 777, 1935; for the application of Muschelisvili's method see A. I. 
Lourye, Bull. Polytech. Inst., Leningrad, vol. 31, p. 305, 1928, and Priklad. Mat. 
Mekhan., vol. 4, p. 93, 1940. See also K. Nasitta, Ingr.-Arch., voi. 23, p. 85, 1955, and 
R. J. Roark, Wisconsin Univ. Eng. Expt. Sta. Bull. 74, 1932. 

2 J. H. Michell, Proc. London Math. Soc., vol. 34, p. 223, 1902. 

3 Several problems of this kind were discussed by A. Nadai, Z. Physik, vol. 23, p. 
066, 1922. Plates supported at several points were also discussed by W. A. Bassali, 
Proc. Cambridge Phil. Soc, vol. 53, p. 728, 1957, and circular plates with mixed bound- 
ary conditions by G. M. L. Gladwell, Quart. J. Mech. Appl. Math., vol. 11, p. 159, 
1958. 




294 THEORY OF PLATES AND SHELLS 



m = l 



in which the summation is extended over all the concentrated reactions (c). 

The general solution of the homogeneous equation (6) is given by expression (195) 
(page 284). Assuming that the plate is solid and omitting the terms that give infinite 
deflections and moments at the center, we obtain from expression (195) 

00 

n>i = A + B r 2 + V (A m r m + C m r m+2 ) cos mO 

w = 1 

00 

+ ^ (A' m r m + C' m r m +^) sin me (e) 
m = l 

For determining the constants we have the following conditions at the boundary: 

(,M r )r- a = ~D 7+v(- ] =0 

^-(*-^)--t56 + 2--*) 



(/) 



m = l 



in which M rt and Q r are given by Eqs. (192) and (193). 

Let us consider a particular case in which the plate is supported at two points which 
are the ends of a diameter. We shall measure 6 from this diameter. Then 71 = 0, 
72 = ?r, and we obtain 

Pa * f , 1 + V / W 2 \ 

w = Wo + w+td ( 2,og2 - J + r^( 210 * 2 -T2) 

00 

- v r i , 2d + ,) _p* 1 i r 

Li [m(m - 1) + (1 - ,)( m - i) m > " m(m + 1) p " C0S m * Cff) 



m = 2,4,0,... 



in which w is the deflection of the simply supported and symmetrically loaded plate, 
P is the total load on the plate, and p = r/a. When the load is applied at the center, 
we obtain from expression (g), by assuming v = 0.25, 

, v Pa 2 
(w)p-o = 0.116 

Pa 2 

Mp-1.0-7r/2 = 0.118 — 

For a uniformly loaded plate we obtain 

(uOp-o = 0.269 ?jjj- 

(w)p-i.0-,t/2 = 0.371 -— 



PLATES OF VARIOUS SHAPED 



295 



By combining two solutions of the type (g), the case shown in Fig. 144 can also be 
obtained. 

When a circular plate is supported at three points 
120° apart, the deflection produced at the center of the 
plate, when the load is applied at the center, is ^_ , 

Fig. 144 



^3E^ 



Mp=o 



Pa* 
0.0670 — 



When the load is uniformly distributed, the deflection at the center is 

Pa 2 
Wp,o = 0.0362 — 

where P = ira 2 q. 

The case of a circular plate supported at three points was investigated by experi- 
ments with glass plates. These experiments showed a very satisfactory agreement 
with the theory. 1 

66. Plates in the Form of a Sector. The general solution developed for circular 

plates (Art. 62) can also be adapted for a plate 
in the form of a sector, the straight edges of 
which are simply supported. 2 Take, as an ex- 
ample, a plate in the form of a semicircle simply 
supported along the diameter AB and uni- 
formly loaded (Fig. 145). The deflection of this 
plate is evidently the same as that of the circular 
plate indicated by the dashed line and loaded as 
shown in Fig. 1456. The distributed load is 
represented in such a case by the series 




* — 

4 

4 

< 

► 4 



(b) 



Q = 



m = 1,3,5, 

and the differential equation of the deflection surface is 



oo 

/ — si 



sin mO 



(a) 



00 

i V ^ ■ a 

AAw = — / sin md 

D Z-/ mir 



(b) 



m = 1,3,5, 



The particular solution of this equation that satisfies the boundary conditions along 
the diameter AB is 



Wo = 



00 



Aqr 4 



7rm(16 - m 2 )(4 - m 2 )D 



sin md 



(c) 



m = 1,3,5, 

The solution of the homogeneous differential equation (194) that satisfies the condi- 

1 These experiments were made by Nadai, ibid. 

2 Problems of this kind were discussed by Nadai, Z. Ver. deuL Ing., vol. 59, p. 169, 
1915. See also B. G. Galerkin, " Collected Papers," vol. 2, p. 320, Moscow, 1953, 
which gives numerical tables for such cases. 



296 THEORY OF PLATES AND SHELLS 

tions along the diameter AB is 

00 

wi = ) (A m r m + B m r m + 2 ) sin mO (d) 

m =1,3,5,... 

Combining expressions (c) and (d), we obtain the complete expression for the deflection 
w of a semicircular plate. The constants A m and B m are determined in each particular 
case from the conditions along the circular boundary of the plate. 
In the case of a simply supported plate we have 

(w)r-« = 

(e) 



[d 2 w (\ dw 1 d*w\l 



Substituting the sum of series (c) and (d) for w in these equations, we obtain the 
following equations for calculating A m and B m : 



A m a™ + B m a»>+* « * qa * 



mw(16 - m 2 )(4 - m 2 )D 
A m a m [m(m - 1) - vm(m - 1)] + B m a m+2 (m + l)[m +2 + ^(2 - m)] 

4qa*[12 + *(4 - m 2 )] 



m7r(16 - ra 2 )(4 - m 2 )D 
From these equations, 

A = qa> 4 (™> + 5 + y) 

a"m7r(16 - m 2 )(2 + m)[m + £(1 + *)]Z) 
p _ ga 4 (m + 3 + v) 



a" +5 W(4 + m)(4 - m 2 )[m + £(1 + „)]£> 



With these values of the constants the expression for the deflection of the plate 
becomes 



00 

V* y (4H 



m7r(16 - ra 2 )(4 - ra 2 ) 
m= 1,3,5, ... 

.r™ m +5 + v 



a m mw(16 - m 2 )(2 + m)[m + £(1 + „)] 



_ r^ m +3 + v ) 

a"+ 2 m^(4 + m)(4 - m 2 )[m + £(1 + „)] J Sm ™* 

With this expression for the deflection, the bending moments are readily obtained 
from Eqs. (192). 

In a similar manner we can obtain the solution for any sector with an angle w/k, k 
being a given integer. The final expressions for the deflections and bending moments 
at a given point can be represented in each particular case by the following formulas: 

qa 4 
W « a — Mr = pqa 2 M t = piqa 2 (/) 



PLATES OF VARIOUS SHAPES 



297 



in which a, /3, and /3i are numerical factors. Several values of these factors for points 
taken on the axis of symmetry of a sector are given in Table 65. 

Table 65. Values of the Factors <*, /3, and 0i for Various Angles w/k 
of a Sector Simply Supported at the Boundary 

v = 0.3 





r/a -'J 


r/a = i 


r/a = 1 


r/a = 1 


ir /k 


a 





0i 


a 





0i 


a 





0i 


a 





01 


ir/4 

7T/3 

7r/2 

7T 


0.00006 
0.00019 
0.00092 
0.00589 


-0.0015 

-0.0025 

0.0036 

0.0692 


. 0093 
0.0177 
0.0319 
0.0357 


0.00033 
0.00080 
0.00225 
0.00811 


0.0069 
0.0149 
0.0353 
0.0868 


0.0183 
0.0255 
0.0352 
0.0515 


0.00049 
0.00092 
0.00203 
0.00560 


0.0161 
0.0243 
0.0381 
0.0617 


0.0169 
0.0213 
0.0286 
0.0468 














0.0025 
0.0044 
0.0088 
0.0221 



The case in which a plate in the form of a sector is clamped along the circular 
boundary and simply supported along the straight edges can be treated by the same 
method of solution as that used in the preceding case. The values of the coefficients 
a and /3 for the points taken along the axis of symmetry of the sector are given in 
Table 66. 

Table 66. Values of the Coefficients a and for Various Angles ir/k 

of a Sector Clamped along the Circular Boundary and Simply 

Supported along the Straight Edges 

v = 0.3 



*/k 


r/i 


»-* 


r/a •■ 


_ i 

- "S 


r/a ■ 


_ 3 
~ 4 


r/a = 1 


a 


P 


a 





a 


P 


or 


p 


7r/4 

7r/3 

7r/2 

7T 


0.00005 
0.00017 
0.00063 
0.00293 


-0.0008 

-0.0006 

0.0068 

0.0473 


0.00026 
0.00057 
0.00132 
0.00337 


0.0087 
0.0143 
0.0272 
0.0446 


0.00028 
0.00047 
0.00082 
0.00153 


0.0107 
0.0123 
0.0113 
0.0016 








-0.0250 
-0.0340 
-0.0488 
-0.0756 



It can be seen that in this case the maximum bending stress occurs at the mid-point 
of the circular edge of the sector. 

If the circular edge of a uniformly loaded plate having the form of a sector is entirely 
free, the maximum deflection occurs at the mid-point of the unsupported circular edge. 
For the case when ir/k = tt/2 we obtain 

qa 4 



Wmax = 0.0633 



D 



The bending moment at the same point is 

M t = 0.1331<?a 2 
In the general case of a plate having the form of a circular sector with radial edges 



298 



THEORY OF PLATES AND SHELLS 



clamped or free, approximate methods must be applied. 1 However, the particular 
problem of a wedge-shaped plate carrying a lateral load can be solved rigorously (see 
Art. 78). Another problem which allows an exact solution is that of bending of a 
plate clamped along two circular arcs. 2 Bipolar coordinates must be introduced in 
that case and data regarding the clamped semicircular plate in particular are given 
in Table 67. 

Table 67. Values of the Factors a, p, and p l [Eqs. (/)] for a Semicircular 
Plate Clamped along the Boundary (Fig. 145a) 
v = 0.3 



Load distribution 



Uniform load q 

Hydrostatic load qy/a 



r/a = 



-0.0731 
-0.0276 



r/a = 0.483 

Pmax 



0.0355 



r/a = 0.486 



0.00202 



r/a = 0.525 

Pl max 



0.0194 



r/a = 1 



-0.0584 
-0.0355 



Bipolar coordinates can also be used to advantage in case of a plate clamped between 
an outer and an inner (eccentric) circle and carrying a single load. 3 

67. Circular Plates of Nonuniform Thickness. Circular plates of nonuniform 
thickness are sometimes encountered in the design of machine parts, such as dia- 
phragms of steam turbines and pistons of reciprocating engines. The thickness of 
such plates is usually a function of the radial distance, and the acting load is sym- 
metrical with respect to the center of the plate. We shall limit our further discussion 
to this symmetrical case. 

Proceeding as explained in Art. 15 and using the notations of that article, from the 
condition of equilibrium of an element as shown in Fig. 28 (page 52) we derive the 
following equation: 



M r + 



dM r 
dr 



r - Mt + Qr = 



(a) 



1 See G. F. Carrier and F. S. Shaw, Proc. Symposia Appl. Math., vol. 3, p. 125, 1950; 
H. D. Conway and M. K. Huang, J. Appl. Mechanics, vol. 19, p. 5, 1952; H. R. Hasse, 
Quart Mech. Appl Math., vol. 3, p. 271, 1950. The case of a concentrated load has 
been discussed by T. Sekiya and A. Saito, Proc. Fourth Japan. Congr. Appl. Mech., 
1954, p. 195. For plates bounded by two radii and two arcs and clamped see G. F. 
Carrier, /. Appl. Mechanics, vol. 11, p. A-134, 1944. The same problem with various 
edge conditions was discussed by L. I. Deverall and C. J. Thome, J. Appl. Mechanics, 
vol. 18, p. 359, 1951. The bending of a uniformly loaded semicircular plate simply 
supported around the curved edge and free along the diameter (a " diaphragm' ' of a 
steam turbine) has been discussed in detail by D. F. Muster and M. A. Sadowsky, 
J. Appl. Mechanics, vol. 23, p. 329, 1956. A similar case, however, with a curved 
edge clamped, has been handled by H. Mliggenburg, Ingr.-Arch., vol. 24, p. 308, 1956. 

2 Green's function for these boundary conditions has been obtained by A. C. Dixon, 
Proc. London Math. Soc, vol. 19, p. 373, 1920. For an interesting limiting case see 
W. R. Dean, Proc. Cambridge Phil. Soc, vol. 49, p. 319, 1953. In handling distributed 
loads the use of the rather cumbersome Green function may be avoided; see S. 
Woinowsky-Krieger, J. Appl. Mechanics, vol. 22, p. 129, 1955, and Ingr.-Arch., vol. 24, 
p. 48, 1956. 

3 This problem was discussed by N. V. Kudriavtzev, Doklady Akad. Nauk S.S.S.R., 
vol. 53, p. 203, 1946. 



PLATES OF VARIOUS SHAPES 299 

in which, as before, 



M r 

M t 



where 



-»(£+ 


■:•) 


-»(! + . 


d<p\ 
dr) 


dw 




v= "* 





(b) 



(c) 



°i 



and Q is the shearing force per unit length of a circular section of radius r. In the case 
of a solid plate, Q is given by the equation 

1 f r 

Q = / q2irrdr id) 

2irr Jo 

in which q is the intensity of the lateral load. 

Substituting expressions (6), (c), and (d) in Eq. (a) and observing that the flexural 
rigidity D is no longer constant but varies with the radial distance r, we obtain the 
following equation: 

(d J , + A + <w(<h + v A i = _i f' qrdr (e) 

\dr r J dr \dr r J r Jo 

Thus the problem of bending of circular symmetrically loaded plates reduces to the 
solution of a differential equation (e) of the second order with variable coefficients. 
To represent the equation in dimensionless form, we introduce the following notations: 

a = outer radius of plate 

h = thickness of plate at any point 

h = thickness of plate at center 

r h 

then - = x ■— = y 

a ho 

We also assume that the load is uniformly distributed. Using the notation 

_ 6(1 - v 2 )a*q 
V " Eh\ 

Eq. (e) then becomes 

^,A + li^j/ 3 \rff_/l_i:^£gl 3 V = _^ (198) 

dx* \x dx ) dx \x 2 x dx ) y l 

In many cases the variation of the plate thickness can be represented with sufficient 
accuracy by the equation 1 

y = e -#* 2 /6 (A) 

in which /3 is a constant that must be chosen in each particular case so as to approximate 
as closely as possible the actual proportions of the plate. The variation of thickness 



(/) 



Or) 



1 



The first investigation of bending of circular plates of nonuniform thickness was 
made by H. Holzer, Z. ges. Turbinenwesen, vol. 15, p. 21, 1918. The results given in 
this article are taken from O. Pichler's doctor's dissertation, "Die Biegung kreissym- 
metrischer Flatten von veranderlicher Dicke/' Berlin, 1928. See also the papex by 
R. Gran Olsson, Ingr.-Arch., vol. 8, p. 81, 1937. 



300 



THEORY OF PLATES AND SHELLS 



along a diameter of a plate corresponding to various values of the constant P is shown 
in Fig. 146. Substituting expression (h) in Eq. (198), we find 



2 + (i-" x ) d f x -(h + vfi )^- pxef,xin 



It can be readily verified that 



<po = — 



V rP**n 



(3 - y)P 



xe p 



(0 



0") 



is a particular solution of Eq. (i). One of the two solutions of the homogeneous equa- 
tion corresponding to Eq. {%) can be taken in the form of a power series: 



<pi 



00 

- „ I", m V £"(1 + y )(3 + «>) • • • (2n - 1 + ,) t .1 
L Zv2-4.4-6-6--.2n- 2n(2n +2) 

71=1 



(*) 



in which ai is an arbitrary constant. The second solution of the same equation 
becomes infinitely large at the center of the plate, i.e., for x = 0, and therefore should 




not be considered in the case of a plate without a hole at the center. If solutions 
(j) and (*) are combined, the general solution of Eq. (i) for a solid plate can be put in 
the following form : 



.,[ 



CVi - 



(3 - jO/9 



*'/2 



(I) 



The constant C in each particular case must be determined from the condition at the 
boundary of the plate. Since series (k) is uniformly convergent, it can be differen- 
tiated, and the expressions for the bending moments can be obtained by substitution 
in Eqs. (6). The deflections can be obtained from Eq. (c). 

In the case of a plate clamped at the edge, the boundary conditions are 



(uO«-i - 
and the constant C in solution (I) is 

C = 



(<p).-i - 
eP/2 



(3 — v)p(<pi) Xm .i 



(m) 



(n) 



To get the numerical value of C for a given value of /?, which defines the shape of the 
diametrical section of the plate (see Fig. 146), the sum of series (Jfc) must be calculated 
for a; = 1. The results of such calculations are given in the above-mentioned paper 
by Pichler. This paper also gives the numerical values for the derivative and for the 



PLATES OF VARIOUS SHAPES 



301 



integral of series (k) by the use of which the moments and the deflections of a plate 
can be calculated. 

The deflection of the plate at the center can be represented by the formula 



Wmax = ocap = a 



6(1 - v 2 )a*q 



(o) 



in which a is a numerical factor depending on the value of the constant 0. Several 
values of this factor, calculated for v = 0.3, are given in the first line of Table 68. 

Table 68. Numerical Factors a and a for Calculating Deflections 
at the Center of Circular Plates of Variable Thickness 

v = 0.3 



OL 

I 
OL 



0.0801 
0.2233 



0.0639 
0.1944 



0.0505 
0.1692 



0.0398 
0.1471 



0.0313 
0.1273 



0.0246 
0.1098 



_o 



0.0192 
0.0937 



-3 



0.0152 
0.0791 



-4 



0.01195 
0.06605 



The maximum bending stresses at various radial distances can be represented by the 
formulas 



( \ 4- ^ 

\<Tr) max = ±7 , 2 
tin 



i \ + 3qa * 



(P) 



The values of the numerical factors y and 71 for various proportions of the plate and 
for various values of x = r/a are given by the curves in Figs. 147 and 148, respectively. 




Fig. 147 

For p = these curves give the same values of stresses as were previously obtained for 
plates of uniform thickness (see Fig. 29, page 56). 

In the case of a plate simply supported along the edge, the boundary conditions are 



(uO«-i = (Mrh-i = 



(Q) 



302 



THEORY OF PLATES AND SHELLS 



Investigation shows that the deflections and maximum stresses can be represented 
again by equations analogous to Eqs. (o) and (p). The notations «', 7', and y x will 
be used for constants in this case, instead of a, 7, and 71 as used for clamped plates. 
The values of a are given in the last line of Table 68, and the values of 7' and y\ are 
represented graphically in Figs. 149 and 150, respectively. 



fimA \ 






















= 0-; 
£= -2'/ 
































/J-4 - 








/J=.-4' 






4[ s "^ — — 


^ur 















/?=2 n 


-a=3 "d 














0=4 - 





0.2! 



Vi 



0.4 



0.2 
Fig. 148 



0.4 0.6 

X >- 



0.8 1.0 




Fig. 150 

To calculate the deflections and stresses in a given plate of variable thickness we 
begin by choosing the proper value for the constant /3 as given by the curves in Fig. 
146. When the value of /3 has been determined and the conditions at the boundary 
are known, we can use the values of Table 68 to calculate the deflection at the center 
and the curves in Figs. 147, 148 or 149, 150 to calculate the maximum stress. If the 
shape of the diametrical section of the given plate cannot be represented with satis- 
factory accuracy by one of the curves in Fig. 146, an approximate method of solving the 
problem can always be used. This method consists in dividing the plate by con- 
centric circles into several rings and using for each ring formulas developed for a ring 



PLATES OF VARIOUS SHAPES 



303 




r ^^ 



plate of constant thickness. The procedure of calculation is then similar to that 

proposed by R. Grammel for calculating stresses in rotating disks. 1 

68. Annular Plates with Linearly 

Varying Thickness. Let us consider a 

circular plate with a concentric hole and 

a thickness varying as shown in Fig. 151. 

The plate carries a uniformly distributed 

surface load q and a line load p = P/2irb 

uniformly distributed along the edge of 

the hole. 2 Letting Do = Eh z Q /12(l - v 2 ) 

be the flexural rigidity of the plate at 

r = b, we have at any distance r from 

the center 

Dor 3 , . 

(a) 




D = 



6 3 



Fig. 151 



Substituting this in Eq. (e) of Art. 67 and taking into account the additional shear 
force P/2irr due to the edge load, we arrive at the differential equation 



dr 2 



+ 4r-^ + (3* - \)<p 
dr 



2D 



-0--)- 



Pb 3 

27rD r 2 



(6) 



The solution of the homogeneous equation corresponding to Eq. (6) is readily 
obtained by setting v = r a . Combining this solution with a particular solution of 
Eq. (6), we get 

qb* qb h P6 3 



V = Ar a i + Br a * + 



+ 



2D (1 - 3v) 6(1 - v)D*r* 6tt(1 - v)D r 2 



in which 



on 



-1.5 + V3.25 - 'iv a, = -1.5 - V3.25 
In the special case v = \, expression (c) has to be replaced by 



3v 



B 

<p = A + - 



qb* , r 

log - 

6Dc b 



qb* 



+ 



Pb* 



4D r 2 47rD r 2 



(c) 



id) 



(e) 



The arbitrary constants A and B must be determined from the respective conditions 
on the boundary of the plate. Writing, for brevity, <p b for (<p) r -&, and M b for (M r ) r -6, 
and introducing likewise <p a , M aj the last column of Table 69 contains the boundary 
conditions and the special values of q and P assumed in six different cases. The same 
table gives the values of coefficients k and hi calculated by means of the solution (e) and 
defined by the following expressions for the numerically largest stress and the largest 
deflection of the plate: 

1 R. Grammel, Dinglers Polytech. J., vol. 338, p. 217, 1923. The analogy between 
the problem of a rotating disk and the problem of lateral bending of a circular plate of 
variable thickness was indicated by L. Foppl, Z. angew. Math. Mech., vol. 2, p. 92, 
1922. Nonsymmetrical bending of circular plates of nonuniform thickness is dis- 
cussed by R. Gran Olsson, Ingr.-Arch., vol. 10, p. 14, 1939. 

2 Thife'ease has been discussed by H. D. Conway, /. Appl. Mechanics, vol. 15, p. 1, 
1948. Numerical results given in Table 69 are taken from that paper. 



304 



THEORY OF PLATES AND SHELLS 



Table 69. Values of Coefficients in Eqs. (/) for Various Values of the 

Ratio a/b (Fig. 151) 















v = 


. l 










Case 

(number 

corresponding 

to Table 3) 


Coeffi- 
cient 


a/6 


Bound- 
! ary 
condi- 
tions 


1.25 


1.5 


2 


3 


4 


5 


3 




k 


0.249 
0.00372 


0.638 
0.0453 


3.96 
0.401 


13.64 
2.12 


26.0 
4.25 


40.6 
6.28 


P - Q* 
Vb = 

M a = 


4 


i 
i 


i 


1 


A; 


0.149 
0.00551 


0.991 
0.0564 


2.23 
0.412 


5.57 
1.673 


7.78 
2.79 


9.16 
3.57 


P = 

<Pb = 

M a = 


5 


Asa 


ft 


0.1275 
0.00105 


0.515 
0.0115 


2.05 
0.0934 


7.97 
0.537 


17.35 
1.261 


30.0 
2.16 


P - Q* 
vt = 
e. = 


6 


D€W 




0.159 
0.00174 


0.396 
0.0112 


1.091 
0.0606 


3.31 
0.261 


6.55 
0.546 


10.78 
0.876 


q =0 

<Pb - 
¥>„ = 


8 


Mid 


Jfc 


0.353 
0.00816 


0.933 
0.0583 


2.63 
0.345 


6.88 
1.358 


11.47 
2.39 


16.51 
3.27 


9 = 

«% - o 

M„ = 


— 


&ia 


Jfc 


0.0785 
3.00092 


0.208 
0.008 


0.52 
0.0495 


1.27 

0.193 

1 


1.94 
0.346 


2.52 

0.482 

i 


P = 

<Pb - 

?. = o 



* Where Q = *-g(a* - 6*). 



PLATES OF VARIOUS SHAPES 



305 



(0V/max — & ,2 
tl 1 



Wr, 



= fc! 



qa A 
Eh\ 



or 



or 



(^r)raax — 1,2 



P 



Wn 



- ki 



Pa 2 
Eh\ 



(/) 



americal results valid for similar plates with constant thickness have been given 

Table 3. 

69. Circular Plates with Linearly Varying Thickness. In discussing the bending 

the circular plate shown in Fig. 152, l we have to consider two portions of the plate 

parately. 

I. The annular area b < r < a. Provided v ¥^ \ y the slope <p = dw/dr again is 

ven by the expression (c) of Art. 68 without, however, its next to last term. 

2. The inner area r <b. Here we 

have dD/dr = 0, and Eq. (e) of Art. 67 is 

reduced to 







h, 



dtipi d<pi 



qr* 




dr s 



+ r _-_ w . -_ 



Pr 
2irDo 



(a) 



where the subscript i refers to the inner 
portion of the plate. The general solu- 
tion of Eq. (a) is 



(b) 
Fig. 152 




iPi = Air H — — 

r 161>o 



Pr 

SwDi 



(2 log r + 1) (6) 



he constants A, J5 in Eq. (c) of Art. 68, and A i} Bi in Eq. (6) above can be obtained 
om the boundary condition 

(*)r- - 
id the conditions of continuity 



(*>»)r-0 = (^ — ^i)r-6 



= (**-?2\ =0 



Tables 70 and 71 give the deflection w ma * and values of bending moments of the plate 
i two cases of loading. To calculate the bending moment at the center in the case 
r a central load P, we may assume a uniform distribution of that load over a small 
rcular area of a radius c. The moment M r = M t at r = then can be expressed 
i the form 



*~ -*•-£(* -£i)+^- 



(c) 



i this formula M is given by Eq. (83), which holds for a supported plate of constant 
lickness; the second term represents the effect of the edge moment; and the third 
irm, due to the nonuniformity of the thickness of the plate, is given by Table 71. 

1 Clamped and simply supported plates of such a shape were discussed by H. Favre, 
ull. Tech. Suisse romande, vol. 75, 1949. Numerical results given below are due 
lbstantially to H. Favre and E. Chabloz, Bull Tech. Suisse romande, vol. 78, 1952. 



306 



THEORY OF PLATES AND SHELLS 



Table 70. Deflections and Bending Moments of Clamped Circular 

Plates Loaded Uniformly (Fig. 152a) 

v = 0.25 



b 


qa* 


Mr = 0ga 2 




Mi — Piqa'- 


t 


a 


r = 


r - b 


r = a 


r = 


r = b 


r = a 




a 











0i 


0i 


0i 


0.2 


0.008 


0.0122 


0.0040 


-0.161 


0.0122 


0.0078 


-0.040 


0.4 


0.042 


0.0332 


0.0007 


-0.156 


0.0332 


0.0157 


-0.039 


0.6 


0.094 


0.0543 


-0.0188 


-0.149 


0.0543 


0.0149 


-0.037 


0.8 


0.148 


0.0709 


-0.0591 


-0.140 


0.0709 


0.0009 


-0.035 


1.0 


0.176 


0.0781 


-0.125 


-0.125 


0.0781 


-0.031 


-0.031 



Table 71. 



Deflections and Bending Moments of Clamped Circular 
Plates under a Central Load (Fig. 1526) 
v = 0.25 



b 


Pa* 


Mr = M t 
r = 


Mr - 


= 0P 


M t = 


= 0.P 


a 


r = b 


r = a 


r = b 


r = a 




a 


* 
Tl 








0i 


0i 


0.2 


0.031 


-0.114 


-0.034 


-0.129 


-0.028 


-0.032 


0.4 


0.093 


-0.051 


-0.040 


-0.112 


-0.034 


-0.028 


0.6 


0.155 


-0.021 


-0.050 


-0.096 


-0.044 


-0.024 


0.8 


0.203 


-0.005 


-0.063 


-0.084 


-0.057 


-0.021 


1.0 


0.224 





-0.080 


-0.080 


-0.020 


-0.020 



*InEq. (c). 

In the case of a highly concentrated load requiring the use of the thick-plate theory, 
the stress at the center of the bottom surface of the plate is given by the expression 



Gmax 



cro + 



6P 7 l 

hi 



(d) 



D = Do M 



in which ao may be calculated by means of expression (97). 

Assuming next a variation of the flexural rigidity of the plate in accordance with 
the law 

r \m 
-) («) 

ao/ 

where ao denotes a length at least equal to the radius of the plate, we arrive in 
general at a slope <p expressible in terms of the hypergeometric function. 1 The par- 
ticular assumption m = 1/v leads, however, to a solution in a closed form. Taking, 
in addition, v = -J we arrive again at a plate with linearly variable thickness. 2 

1 R. Gran Olsson, Ingr.-Arch., vol. 8, p. 270, 1937. 

2 See especially H. D. Conway, J. Appl. Mechanics, vol. 18, p. 140, 1951, and vol. 20, 
p. 564, 1953. 



PLATES OF VARIOUS SHAPES 



307 



Symmetrical deformation of plates such as shown in Fig. 153 also can be 
;ated by means of a parameter method akin to that described in Art. 39 
mmerical results 1 obtained in that way 
xe given in Tables 72 and 73. 

For bending moments and tensile 
tresses under central load P (Fig. 1536) 
ixpressions 



investi- 
Some 



•~q 



J£ 



h, 



M„ 



ind 



»x = <r« + 



M n + yi P 

6P T 2 



h\ 



(/) 
(9) 



— 



—J 



(a) 



-H 




K-2c 



,£>/ 



(b) 



malogous to Eqs. (c) and (d) may be 
lsed. Mo again is given by expression 
83), ao denotes the value calculated by 
neans of expression (96), and 72 is given 
n Table 73. 

Of practical interest is also a combina- 
tion of loadings shown in Fig. 153a and 
>. Taking q = -P/Va 2 , we have the 
itate of equilibrium of a circular footing 
carrying a central load P and submitted 
it the same time to a uniformly distrib- 
lted soil reaction (Fig. 153c). Some 
lata regarding this case, in particular 
;he values of the factor 72, to be used in 
brmulas (/) and (</), are given in Table 

r4. 2 

Table 72. Deflections and Bending Moments of Simply Supported 

Plates under Uniform Load (Fig. 153a) 

v = 0.25 



hi 



2STf 




ho 
hi 


qa 4 
Wm " = a Ehl 

a 


M T = 

*>* 


Pqa* 


Mi = 0i<?a 2 


r = 



r = a/2 



r - 

/Si- 


r = a/2 
0i 


r = a 
0i 


1.00 


0.738 


0.203 


0.152 


ck 203 


0.176 


0.094 


1.50 


1.26 


0.257 


0.176 


0.257 


0.173 


0.054 


2.33 


2.04 


0.304 


0.195 


0.304 


0.167 


0.029 



1 Due, as well as the method itself, to H. Favre and E. Chabloz, Z. angew. Math. u. 
°hys., vol. 1, p. 317, 1950, and Bull. Tech. Suisse romande, vol. 78, 1952. 

2 For further results concerning circular plates with varying thickness see W. Gittle- 
nan, Aircraft Eng., vol. 22, p. 224, 1950, and J. Paschoud, Schweiz. Arch., vol. 17, p. 
105, 1951. A graphical method of design has been given by P. F. Chenea and P. M. 
Naghdi, J. Appl. Mechanics } vol. 19, p. 561, 1952. 



308 



THEORY OF PLATES AND SHELLS 



Table 73. Deflections and Bending Moments of Simply Supported 
Circular Plates under Central Load (Fig. 1536) 

v = 0.25 



ho 
hi 


Pa 2 
a 


Mr - Mt 


Mr = fiP 


M,= 


■0iP 


r =0 

72 


r = a/2 


r ■» c/2 
/Si 


r — a 
Pi 


1.00 
1.50 
2.33 


0.582 

0.93 

1.39 




0.029 

0.059 


0.069 
0.088 
0.102 


0.129 
0.123 
0.116 


0.060 
0.033 
0.016 



Table 74. Bending Moments of a Circular Footing Plate with Central 
Load and Uniformly Distributed Soil Pressure (Fig. 153c) 

v - 0.25 





Mr = Mt 


Mr = pP 


M t = 


-PiP 


ho 
hi 










r = 


r = a/2 


r = a/2 


r ■« a 




78 


P 


Pi 


/3i 


1.00 


-0.065 


0.021 


0.073 


0.030 


1.50 


-0.053 


0.032 


0.068 


0.016 


2.33 


-0.038 


0.040 


0.063 


0.007 



70. Nonlinear Problems in Bending of Circular Plates. From the 
theory of bending of bars it is known that, if the conditions at the sup- 
ports of a bar or the loading condi- 
tions are changing with the deflection 
of the bar, this deflection will no 
longer be proportional to the load, 
and the principle of superposition 
cannot be applied. 1 Similar prob- 
lems are also encountered in the case 
of bending of plates. 2 A simple ex- 
ample of this kind is shown in Fig. 
154. A circular plate of radius a is 




(b) 
Fig. 154 



pressed by a uniform load q against an absolutely rigid horizontal founda- 
tion. If moments of an intensity M a are applied along the edge of the plate, 
a ring-shaped portion of the plate may be bent as shown in the figure, 

1 An example of such problems is discussed in S. Timoshenko, "Strength of Mate- 
rials," part II, 3d ed., p. 69, 1956. 

2 See K. Girkmann, Stahlbau, vol. 18, 1931. Several examples of such problems are 
discussed also in a paper by R. Hofmann, Z. angew. Math. Mech., vol. 18, p. 226, 1938. 



PLATES OF VARIOUS SHAPES 309 

vhereas a middle portion of radius b may remain flat. Such conditions 
>revail, for example, in the bending of the bottom plate of a circular 
ylindrical container filled with liquid. The moments M a represent in 
his case the action of the cylindrical wall of the container, which under- 
;oes a local bending at the bottom. Applying to the ring-shaped portion 
if the bottom plate the known solution for a uniformly loaded circular 
)late [see expressions (ra) in Art. 62], we obtain the deflection 

w = d + C 2 log r + CV 2 + C^ log r + ^ (a) 

<"or determining the constants of integration Ci, . . . , Ca we have the 
ollowing boundary conditions at the outer edge: 

(W)r-a = (M r ) r=a = -M fl (6) 

Uong the circle of radius b the deflection and the slope are zero. The 
>ending moment M r also must be zero along this circle, since the inner 
>ortion of the plate remains flat. Hence the conditions at the circle of 
adius b are 

(w) r=6 = (^) = (M r )r= 6 = (c) 



3y applying conditions (b) and (c) to expression (a) we obtain the five 
ollowing equations: 

Ci + C 2 log a + C 3 a* + da* log a = - ^ 
Ci + C 2 log b + C 3 b 2 + C 4 6 2 log 6 = - ^ 
~ "~- + C 3 2(v+ 1) 

an 2 M 

+ C 4 (3 + 2 log a + 2, log a + v) = - ^ (3 + ») + ^ (d) 
- + C 3 2(p + 1) 



^2 



a z 



b 



+ C 4 (3 + 2 log 6 + 2? log b + i0 = - ^ (3 + v) 



3 



C*\ + C 3 26 + C 4 6(2 log 6 + 1) = - ^*L 

By eliminating the constants Ci, . . . , C 4 from these equations we obtain 
in equation connecting M a and the ratio 6/a, from which the radius b of 
;he flat portion of the plate can be calculated for each given value of M a . 
With this value of b the constants of integration can be evaluated, and 
:he expression for the deflection of the plate can be obtained from Eq. (a). 
Representing the moment M a and the angle of rotation <p a of the edge of 



310 



THEORY OF PLATES AND SHELLS 



the plate by the equations 



and 



<Pa = 



qa 3 
32D 



W 



/* 



and repeating the above-mentioned calculations for several values of the 
moment M a , we can represent the relation between the constant factors 
a and graphically, as shown in Fig. 155, for the particular case 1 v = 0. 

It is seen from this figure that does not 
vary in proportion to a and that the resist- 
ance to rotation of the edge of the plate 
decreases as the ratio b/a decreases. This 
condition holds up to the value a = 5, at 
which value /3 = 1, b/a = 0, and the plate 
touches the foundation only at the center, 
as shown in Fig. 1546. For larger values 
of a, that is, for moments larger than 
M a = 5qa 2 /S2, the plate does not touch 
the foundation, and the relation between 
a and /S is represented by the straight line 
AB. The value M a = 5qa 2 /S2 is that 
value at which the deflection at the center 
of the plate produced by the moments M a 
is numerically equal to the deflection of a uniformly loaded plate simply 
supported along the edge [see Eq. (68)]. 

Another example of the same kind is shown in Fig. 156. A uniformly 
loaded circular plate is simply supported along the edge and rests at the 
center upon an absolutely rigid foundation. Again the ring-shaped por- 
tion of the plate with outer radius a and inner radius b can be treated as 



1.4 
1.2 
1.0 
0£ 

> 

0.6 
0.4 

0.2 














f 












L 








&■« 


01 J 


\ 










t 


\i 




b-L. 








a 












<<--•; 


h°-t 











2 


3 


4 


5 


6 



cc 

Fig. 155 



m 




" -a ->l*b->n y 

Fig. 156 Fig. 157 

a uniformly loaded plate, and solution (a) can be used. The ratio b/a 
depends on the deflection 5 and the intensity of the load q. 

71. Elliptical Plates. Uniformly Loaded Elliptical Plate with a Clamped 
Edge. Taking the coordinates as shown in Fig. 157, the equation of the 
boundary of the plate is 



x 2 . y* 



-, + h,-i = o 



a 



b 2 



(a) 



1 This case is discussed in the paper by Hofniann, op. oil. 



PLATES OF VARIOUS SHAPES 311 

'he differential equation 

AAw = ^ (6) 

nd the boundary conditions for the clamped edge, i.e., 

w = and — = (c) 

dn 

re satisfied by taking for the deflection w the expression 1 

t is noted that this expression and its first derivatives with respect to 
: and y vanish at the boundary by virtue of Eq. (a). Substituting expres- 
ion (d) in Eq. (6), we see that the equation is also satisfied provided 

Wo = —7 ^ r- (199) 

2)^ + 24 i6_\ 
\a 4 fe 4 a 2 6 2 / 

Thus, since expression (d) satisfies Eq. (6) and the boundary conditions, 
t represents the rigorous solution for a uniformly loaded elliptical plate 
vith a clamped edge. Substituting x = y = in expression (d), we find 
:hat Wo, as given by Eq. (199), is the deflection of the plate at the center, 
[f a = b, we obtain for the deflection the value previously derived for a 
damped circular plate [Eq. (62), page 55]. If a = «>, the deflection w 
becomes equal to the deflection of a uniformly loaded strip with clamped 
snds and having the span 26. 

The bending and twisting moments are obtained by substituting expres- 
sion (d) in Eqs. (101) and (102). In this way we find 

, / x 2 , 3«/ 2 lVI , , 

For the center of the plate and for the ends of the horizontal axis we 
obtain, respectively, 

(A/*)—o.i^.o = 4wqD(-. i + ~j and (M x ) x = a . y =o = ^~ (/) 

1 This solution and the solution for a uniformly varying load q are obtained by 
G. H. Bryan; s,ee A. E. H. Love's book, "Theory of Elasticity," 4th ed., p. 484. The 
case of an elliptical plate of variable thickness is discussed by R. Gran Olsson, Ingr.- 
Arch., vol. 9, p. 108, 1938. 



312 THEORY OF PLATES AND SHELLS 

Similarly, for the moments M y at the center and at the ends of the vertical 
axis we find, respectively, 

(M y ) x =o,v=o = ±WoD (^ + ^J and (M y ) x = ,y=b = 1^_ ( g ) 

It is seen that the maximum bending stress is obtained at the ends of 
the shorter principal axis of the ellipse. Having the moments M XJ M y , 
and M xy , the values of the bending moment M n and the twisting moment 
M nt at any point on the boundary are obtained from Eqs. (c) (Art. 22, 
page 87) by substituting in these equations 

dy b 2 x . dx a 2 y /7 . 

cos a = -j- = — , = sm a = — r = — ; — * __= (h) 

ds \/a A y 2 + b A x 2 ds VaV + b A x 2 

The shearing forces Q x and Q y at any point are obtained by substi- 
tuting expression (<2) in Eqs. (106) and (107). At the boundary the 
shearing force Q n is obtained from Eq. (d) (Art. 22, page 87), and the 
reaction V n from Eq. (g) of the same article. In this manner we find 
that the intensity of the reaction is a maximum at the ends of the minor 
axis of the ellipse and that its absolute value is 

nr\ a 2 b(3a 2 + b 2 )q ^ h 

(7w)ma * = 3a* + 36* + 2a 2 b 2 f ° r a>b W 

The smallest absolute value of V n is at the ends of the major axis of the 
ellipse where 

(v \ = a&2(a2 ± 3b2) g (i\ 

For a circle, a = 6, and we find (F n ) max = (V n ) mln = ga/2. 

Elliptical Plate with a Clamped Edge and Bent by a Linearly Varying Pressure. 
Assuming that q = q x, we find that Eq. (6) and the boundary conditions (c) are 
satisfied by taking 






W - fe 5 1 V (200) 

1 1 

a 4 6 4 a 2 6 2 

From this expression the bending moments and the reactions at the boundary can be 
calculated as in the previous 

Uniformly Loaded Elliptical Plate with Simply Supported Edge. The solution for 
this case is more complicated than in the case of clamped edges; 1 therefore we give here 
only some final numerical results. Assuming that a/b > 1, we represent the deflection 
and the bending moments at the center by the formulas 

qb 4 
Mx-y-o - ct — M,= Pqb* My - p iq b 2 *) 

1 See B. G. Galerkin, Z. angew. Math. Mech., vol. 3, p. 113, 1923. 



PLATES OF VARIOUS SHAPES 



313 



The values of the constant factors a, 0, and 0i for various values of the ratio a/b and 
for v = 0.3 are given in Table 75. 

Table 75. Factors a, 0, Pi in Formulas (A) for Uniformly Loaded and 
Simply Supported Elliptical Plates 

v .,« 0.3 



a/6 


1 


1.1 


1.2 


1.3 


1.4 


1.5 


2 


3 


4 


5 


00 


01 


0.70 

0.206 

0.206 


0.83 

0.215 

0.235 


0.96 

0.219 

0.261 


1.07 

0.223 

0.282 


1.17 

0.223 

0.303 


1,26 

0.222 

0.321 


1.58 

0.210 

0.379 


1.88 

0.188 

0.433 


2.02 

0.184 

0.465 


2.10 

0.170 

0.480 


2.28 

0.150 

0.500 



Comparison of these numerical values with those previously obtained for rectangular 
plates (Table 8, page 120) shows that, for equal values of the ratio of the sides of 
rectangular plates and the ratio a Jb of the semiaxes of elliptical plates, the values of 
the deflections and the moments at the center in the two 
kinds of plate do not differ appreciably. The case of a 
plate having the form of half an ellipse bounded by the 
transverse axis has also been discussed. 1 



72. Triangular Plates. Equilateral Triangular 
Plate Simply Supported at the Edges. The bend- 
ing of such a triangular plate by moments M n 
uniformly distributed along the boundary has 
already been discussed (see page 94). It was 
shown that in such a case the deflection surface of the plate is the same 
as that of a uniformly stretched and uniformly loaded membrane and is 
represented by the equation 




w 



~ 4aD[ X 



Sy*x - a(x* + y 2 ) + 



H 



(a) 



in which a denotes the height of the triangle, and the coordinate axes 
are taken as shown in Fig. 158. 

In the case of a uniformly loaded plate the deflection surface is 2 



w = 



64aZ> 



\z* - Sy 2 x - a(x 2 + V 2 ) + ^ a 3 ] (| a 2 - z 2 - y*\ 



(201) 



1 B. G. Galerkin, Messenger Math., vol. 52, p. 99, 1923. For bending of clamped 
elliptical plates by concentrated forces see H. Happel, Math. Z. y vol. 6, p. 203, 1920, 
and C. L. Perry, Proc. Symposia Appl. Math., vol. 3, p. 131, 1950. See also H. M. 
Sengupta, Bull. Calcutta Math. Soc, vol. 41, p. 163, 1949, and vol. 43, p. 123, 1950; 
this latter paper also contains a correction to the former one. By means of curvi- 
linear coordinates, solutions for plates clamped along some other contour lines and 
submitted to a uniform load have been obtained by B. Sen, Phil. Mag., vol. 33, p. 294, 
1942. 

2 The problem of bending of a plate having the form of an equilateral triangle was 
solved by S. Woinowsky-Krieger, Ingr.-Arch., vol. 4, p. 254, 1933. 



314 THEORY OF PLATES AND SHELLS 

By differentiation we find 



Ate; = — 



4aZ> 



[- 



Sy*x - a(x 2 + y*) + 



H 



Q>) 



It may be seen from (201) and (6) that the deflection and the bending 
moment at the boundary vanish, since the expression in the brackets is 
zero at the boundary. Further differentiation gives 



AAw = -^ 



(c) 



Hence the differential equation of the deflection surface is also satisfied, 
and expression (201) represents the solution of the problem. Having the 
expression for deflections, the expressions for the bending moments and 
the shearing forces can be readily obtained. The maximum bending 
moment occurs on the lines bisecting the angles of the triangle. Con- 
sidering the points along the x axis and taking v = 0.3, we find 



< — 


--CJ. 






-PJ 

/ 
^ / 




f+p 






\ 






\ 

-p; 


» 




/ 




A 


r 




a 


\ 
\ 

^\ 

-P) 

/ 


4- 

> 






1 



(M x ) max = 0.0248ga 2 
(M y ) m = 0.0259ga 2 

At the center of the plate 



at x = -0.062a 
at x = 0.129a 



M x = M y = (1 + v) 



54 



(202) 



(203) 



The case of a concentrated force acting on the 
plate can be solved by using the method of images (see 
page 156). Let us take a case in which the point of 
application of the load is at the center A of the plate 
(Fig. 15Q) . Considering the plate, shown in the figure 
by the heavy lines, as a portion of an infinitely long 
rectangular plate of width a, we apply the fictitious 
loads P with alternating signs as shown in the figure. 
The nodal lines of the deflection surface, produced by 
such loading, evidently divide the infinitely long plate 
into equilateral triangles each of which is in exactly the same condition 
as the given plate. Thus our problem is reduced to that of bending of 
an infinitely long rectangular plate loaded by the two rows of equidistant 
loads +P and —P. Knowing the solution for one concentrated force 
(see Art. 36) and using the method of superposition, the deflection at 
point A and the stresses near that point can be readily calculated, since 
the effect of the fictitious forces on bending decreases rapidly as their dis- 
tance from point A increases. In this manner we find the deflection at A : 



Fig. 159 



wo = 0.00575 



Pa 2 
D 



(204) 



PLATES OF VARIOUS SHAPES 



315 



The bending moments at a small distance c from A are given by the 
expressions 

(1 + OP f log «v5_ 0.3^ _(L=_Of 

" x .. "_ (205) 

V3 



M x = 
M v 



4ir 
(1 + v)l- 



-(*'-. 



0.379 + 



) 



(1 ~ v)P 



4tt \ w irC 

Since for a simply supported and centrally loaded circular plate of radius 
a the radial and the tangential moments at a distance c from the center 
are, respectively (see page 68), 

d + * P log * 

I! + I/ir . fl.n 11 — V\l~ 

and 



Mr = 



47T C 47T 



it can be concluded that the first terms on the right-hand side of Eqs. 
(205) are identical with the logarithmical terms for a circular plate with 
a radius 

a V3 



ao = 



-0.379 



(e) 



Hence the local stresses near the point of application of the load can be 
calculated by using the thick-plate theory developed for circular plates 
(see Art. 19). 

Equilateral Triangular Plates with Two or Three Edges Clamped. Triangular plates 
are used sometimes as bottom slabs of 
bunkers and silos. In such a case each 
triangular plate is rigidly clamped along 
both its inclined edges and clamped 
elastically along its third, horizontal 
edge (Fig. 160). Only the uniform and 
the hydrostatic distribution of the load 
is of practical interest. The largest 
bending moment of the panel and the 
clamping moments at the middle of a 
built-in edge may be represented as 

M = Pqa 2 or M = Piqoa 2 (/) 

according to the type of loading (Fig. Fig. 160 

160). The values of coefficients & and 

Px, obtained by the method of finite differences, 1 are given in Table 76. 

It should be noted, finally, that a plate in form of a triangle with angles tt/2, tt/3, 
and 7r/6 and having all edges simply supported can be considered as one-half of the 
equilateral plate (Fig. 158), this latter being loaded antisymmetrically above the axis 

1 See A. Smotrov, "Solutions for Plates Loaded According to the Law of Trapeze/' 
Moscow, 1936. 




316 



THEORY OF PLATES AND SHELLS 



Table 76. Values of the Factors 0, fix in Eqs. (/) for Equilateral 

Triangular Plates (Fig. 160) 

v = 0.20 



Load distribution 


Edgey — simply supported 


Edge y = clamped 


M.i 


M vl 


Mni 


M vi 


M xl 


M vl 


Afn2 


M v » 


Uniform p 

Hydrostatic 0i 


0.0126 
0.0053 


0.0147 
0.0035 


-0.0285 
-0.0100 






0.0113 
0.0051 


0.0110 
0.0034 


-0.0238 
-0.0091 


-0.0238 
-0.0060 



x. The problem of bending of such a plate can be solved in several ways— for example, 

by the method of images. 1 

Plate in the Form of an Isosceles Right Triangle with Simply Supported Edges. Such 

a plate may be considered as one-half of a square 
plate, as indicated in Fig. 161 by dashed lines, 
and the methods previously developed for rectan- 
gular plates can be applied. 2 If a load P is 
applied at a point A with coordinates £, rj (Fig. 
161), we assume a fictitious load — P applied at 
A', which is the image of the point A with respect 
to the line BC. These two loads evidently pro- 
duce a deflection of the square plate such that the 
diagonal BC becomes a nodal line. Thus the 
portion OBC of the square plate is in exactly the 
same condition as a simply supported triangular 
plate OBC. Considering the load +P and using 

the Navier solution for a square plate (page 111), we obtain the deflection 




W\ = 



4Pa* 
r 4 Z> 



m= 1 ri=* 1 



. mx£ . mrrj 

sin sin — 

a a . ntirx . nwy 

— . _ ■ — — — sin sin 

(m 2 + n 2 ) 2 a a 



(g) 



In the same manner, considering the load 
instead of tj, we obtain 



-P and taking a — rj instead of £ and a — £ 



w* = — 



4Po 2 






^hu- 



i) 



m+n 



. mvrrj . nxf 

sin sin 

a a 

"^m^T^ 2- 



. mirx . niry ._ N 

sin sin (ft) 

a a 



m = l 



The complete deflection of the triangular plate is obtained by summing up expressions 
(g) and (ft): 

w = wi + w 2 (i) 

1 For the solution of this problem in double series, see R. Girtler, Sitzber. Akad. 
Wiss. Wien, vol. 145, p. 61, 1936. The bending of equilateral triangular plates with 
variable thickness has been discussed by H. Gottlicher, Ingr.-Arch., vol. 9, p. 12, 1938. 

2 This method of solution was given by A. Nddai, "Elastische Platten," p. 178, 
1925. Another way of handling the same problem was given by B. G. Galerkin, Bull, 
acad. sci. Russ., p. 223, 1919, and Bull. Poly tech. Inst., vol. 28, p. 1, St. Petersburg, 
1919. 



PLATES OF VARIOUS SHAPES 



317 



To obtain the deflection of the triangular plate produced by a uniformly distributed 
load of intensity q, we substitute q d£ drj for P and integrate expression (t) over the 
area of the triangle OBC. In this manner we obtain 



w 



i6«a_ 4 y y 

m = 1,3,5,... n = 2,4,6, . 



. mirx . rnry 

n sm sin 

a a 

m(n* — ra 2 )(ra 2 + n 2 ) 2 



+ 



XT\ X A m sin sm — r 

X X ?L__fL_ 

/ v / |/ n(m 2 - n 2 )(ra 2 + n 2 ) 2 



(i) 



m= 2,4,6, ... n = l,3,5, ... 



This is a rapidly converging series and can be used to calculate the deflection and the 
bending moments at any point of the plate. Taking the axis of symmetry of the 
triangle in Fig. 161 as the xi axis and representing the deflections and the moments 
M Xl and M Vl along this axis by the formulas 



qa 4 



M Vl = Piqa* 



(*) 



the values of the numerical factors a, p, and /3i are as given in Figs. 162 and 163. By 
comparing these results with those given in Table 8 for a uniformly loaded square 




Fig. 162 



-0.0134 
-0.0100 

-0.0050 - 




Fig. 163 



318 THEORY OF PLATES AND SHELLS 

plate, it can be concluded that for the same value of a the maximum bending moment 
for a triangular plate is somewhat less than half the maximum bending moment for a 
square plate. 

To simplify the calculation of the deflections and moments, the double series (j) can 
1 be transformed into simple series. 1 For this purpose we use the known series 

Scosh ml- — x I 
cos nx _ _2_ _^r \2 / 
(n 2 + m 2 ) 2 ~" ~ m A 2w» . , irm 
sinh 



U m (x) 



o a ii sinh — 

n = 2,4,6, ... 2 



(i-) 



sinh m \ 

ir 2 cosh mx irx N - jr_ 

4m 2 . , irm 2m 2 . mw 

sinh 2 — sinh — 

2 2 

which can be represented in the following form 

2 

Um(x) = (« m + Pmx) cosh ?nx + (7m + 5 m x) sinh rax (m) 

Considering now the series 



we obtain 



00 

- 2 

n = 2,4,fi,. 



^ cos nx 

Vm{x) = ' - (n 2 + w*)«(n* - m 2 ) (n) 



dF m \ n sin ?ix 



- s 



dx jLi (n 2 + m 2 ) 2 (n 2 - ra 2 ) 

n = 2,4,0,... 



»* > 

I: 



\ dx Jx-Q 



(o) 



j dVm , 2 T7 \ cosnx 

and — - — + m 2 V m = - / — — — = -£/«■ (p) 

dx 2 Lj (n 2 + m 2 ) 2 

n = 2,4,0, . . . 
By integrating Eq. (p) we find 

V m = ^4m cos mx + B m sin mx + — I U m (£) sin ?n(£ - x) d£ (q) 



dV 

and —r-^ = — mA m sin ?>?x + mB m cos mx — / U m (g) cos m(£ — x) d{ (r) 

dx 

The constants A m and J3 W can be determined from the conditions 

and V m (0) = V m (w) (s) 



which follow from series (o) and (n). With these values of the constants expression 
(r) gives the sum of series (o), which reduces the double series in expression (j) to a 
simple series. 

73. Skewed Plates. Plates bounded by an oblique parallelogram have been used 
recently as floor slabs of skew bridges. Such slabs usually are simply supported along 

1 This transformation was communicated to S. Timoshenko by J. V. Uspensky. 



PLATES OF VARIOUS SHAPES 



319 



o 



--ma- 



the abutments, whereas both other sides remain free or are supported efastically by 
"curbs" or beams. 

In the most general case the use of an oblique system of coordinates chosen in 
accordance with the given angle of skew 
should be recommended ; in certain particular 
cases rectangular coordinates may also be used 
to advantage in dealing with skew plates, and 
the method of finite differences appears, in 
general, to be the most promising. The fol- 
lowing numerical data for uniformly loaded 
skewed plates were obtained in that way. 1 At 
the center of a skew plate with all edges simply 
supported (Fig. 164a), let Y \ 



v Supp.- 



<-Supp.-- x 



\ 




W = a 



qa A 
~D 



iWmax = 0ga 2 



(a) 



(o) 



V 



The bending moment M mrvx acts very nearly 
in the direction of the short span of the plate. 
If the edges y = and y = a are free and 
the other two edges are simply supported 
(Fig. 1646), the central portion of the plate 
carries the load in the direction normal to the 
abutments. Letting wo and (Mo) max be, 
respectively, the deflection and bending 
moment at the center of the plate, and (wi) ma x 
and (ilfi)max the corresponding quantities at 
the free edge, we may express these quantities in the form 




Fig. 164 



(I a 4 
Wo = CKO — (Jlfo)max = PoQCi 2 



(Wl)max = Oil 



qa 4 



<M 



(Mi) max = Piqa 2 



The numerical values of the coefficients are given in Table 77. 

74. Stress Distribution around Holes. In order to investigate the 
stress distribution around a hole, it is simplest to consider a very large 
plate ; results obtained in this way prove to be applicable without appreci- 
able inaccuracy to plates of any shape, provided the width of the hole 
remains small as compared with the over-all dimensions of the plate. 

1 The most data are due to V. P. Jensen, Univ. Illinois Bull 332, 1941, and V. P- 
Jensen and J. W. Allen, Univ. Illinois Bull 369, 1947. See also C. P. Siess, Proc 
ASCE, vol. 74, p. 323, 1948. Analytical methods have been applied by H. Favre, 
Schweiz. Bauztg., vol. 60, p. 35, 1942; P. Lardy, Schweiz. Bauztg., vol. 67, p. 207, 1949; 
and also by J. Krettner, Ingr.-Arch., vol. 22, p. 47, 1954, where further bibliography 
is given. For use of energy methods see also A. M. Guzman and C. J. Luisoni, 
Pubis. Univ. Nad. Buenos Aires, p. 452, 1953. Pure bending of skewed plates has 
been discussed by E. Reissner, Quart. Appl. Math., vol. 10, p. 395, 1953. Models of 
skewed plates were tested by L. Schmerber, G. Brandes, and H. Schambeck, Bauinge- 
nieur, vol. 33, p. 174, 1958. For use of finite differences see also Art. 83. 



320 



THEORY OF PLATES AND SHELLS 



Table 77. Values of Coefficients in Eqs. (a) and (b) for Deflections 
and Bending Moments of Uniformly Loaded Skewed Plates 

v = 0.2 





m 


n 


Plate in Fig. 164a 


Plate in Fig. 1646 


¥>" 


a 





«0 


«i 


/So 


Pi 



30 
30 
45 
60 
75 


2 

2.02 

1.92 

2 

2 

2 


2 

1.75 

1.67 

1.414 

1 

0.518 


0.01013 
0.01046 

0.00938 
0.00796 
0.00094 


0.0999 
0.0968 

0.0898 
0.0772 
0.0335 


0.214 

0.1183 
0.0708 
0.0186 


0.224 

0.1302 
0.0869 
0.0396 


0.495 

0.368 
0.291 
0.166 


0.508 

0.367 
0.296 
0.152 



To take an example, let us consider an infinitely large plate in a uni- 
form state of stress denned by the bending moments 

M' x = M M' y = (a) 

which correspond to a deflection surface 

M r 2 



w 



, _ M (x 2 - vy 2 ) = 

' ~ 2D(1 - v 2 ) 4Z>(1 



v 2 ) 



[1 - v + (1 + v) cos 20] (b) 



To obtain the disturbance produced in such a state of pure bending 

by a circular hole with a radius a (Fig. 
165), we assume the material to be re- 
moved inside the periphery of the circle. 
Then we have to replace the action of 
the initial stresses along the periphery of 
the hole by the action of the external 
couples and forces: 




Fig. 165 



{M' T )r-a = ^ (1 + cos 26) 

(V' r ) r -a = ^ COS 26 



(c) 



which are readily obtained by differentiation of expression (6) in accord- 
ance with Eqs. (192). 

On the initial state of stress we superimpose now an additional state 
of stress such that (1) the combined couples and forces vanish at r = o 
and (2) the superimposed stresses taken alone vanish at infinity (r = «). 

We can fulfill both conditions by choosing the additional deflection in 
the form 

^[Alogr + (B + C^)cos20] (d) 



w" = 



PLATES OF VARIOUS SHAPES 321 

This expression also satisfies the homogeneous differential equation (194) 
md yields the following stress resultants on the periphery of the hole: 

W),-. = - ^"° {(1 - *)A + [4vB - 6(1 - v)C] cos 20} 

M (e) 

(y;')r=a = — [(6 - 2v)A + 6(1 - v)C] cos 2d 
T a 

Since expressions (c) and (e) for M r contain a constant term as well as 
i term proportional to cos 26, while both expressions for V r contain only 
me term, three equations are needed to satisfy the required conditions 
/jf > _|_ m " = and V' r + V" = on the periphery of the hole. Resolv- 
ng these equations with respect to the unknown coefficients A, B, and C, 
ive obtain the final deflections w = w' + w" and the following stress 
resultants along the periphery of the plate: 

M t = M \l - 2( Q 1 ]j" ,,) cos2gl 

L 3 + v j (/) 

Qt = 4Mo _ sin 2e 
* (3 + y)a 

For 6 = 7r/2 and 6 = tt/4, respectively, we obtain 

(M«W = |^> ^ 

(Q ' )m " = (3 + F)a M ° 

It is usual to represent the largest value of a stress component due to a 
local disturbance in the form 

where a denotes the average value of the respective component in the 
same section and A; is the so-called factor of stress concentration. Having 
in mind the largest bending stress along the periphery of the hole, 
we can also write fc = (Af,)m„/Afo, M being the initial value of the stress 
couples at 6 = x/2, where this largest stress occurs. Thus in the event 
of pure bending we have 

k = ^±^ (i) 

3 + v 

equal to about 1.80 for steel (v = £). 

Factors of stress concentration could be obtained in a similar manner 
for various modes of a uniform state of stress and also for holes of other 
than circular shape. 1 All such results, however, prove to be of relatively 
little value for the following reason. 

1 See J. N. Goodier, Phil Mag,, vol. 22, p. 69, 1936, and G. N. Savin, "Stress Con- 
centration around Holes/ ' Moscow, 1951. 



322 



THEORY OF PLATES AND SHELLS 



While the bending stresses (to take only the previously discussed 
case) do not exceed the value of <r max = QM ok/h*, the largest value of the 
corresponding shearing stresses is given by 



_ = 1/) _ 6-frfo _ Cmax h 

W ^ V— - (3 + v)ah - (3 + p)k - 



U) 



Thus, by decreasing the ratio a/h we can increase the ratio w/ir™, at 
will. In this way we soon arrive at transverse shearing stresses of such 
a magnitude that their effect on the plate deformation ceases to be negli- 
gible in comparison with the effect of the couples. Consequently, to 




--Customary plote theory 



"73 



a J 

h 

Fig. 166 

assure reliable results regarding the stress distribution around holes, we 
have to resort to special theories which take the shear deformation into 
account. 

Stress-concentration factors obtained 1 by means of E. Reissner's theory 
(see Art. 39) are plotted in Fig. 166 versus the value of a/h. The curve 
k b holds in the case of pure bending considered above; the curve k t gives 
the stress concentration in the event of a uniform twist, produced by 
couples M x = M , M y = — M in the initial state of stress. The values 

1 E. Reissner, J. Appl. Mechanics, vol. 12, p. A-75, 1945. The case is discussed most 
rigorously by J. B. Alblas, "Theorie van de driedimensionale Spanningstoestand in 
een doorborde plaat," Amsterdam, 1957. For bending of a square plate with a circular 
hole, see M. El-Hashimy, "Ausgewahlte Plattenprobleme," Zurich, 1956, where 
customary theory is applied. 



PLATES OF VARIOUS SHAPES 



323 



fe = 1.80 and k t = 1.60 given for these cases by the customary theory 
ppear, if plotted, as straight lines which approach both respective curves 
asymptotically as the ratio a/h increases indefinitely. It is seen from 
he graph that even for holes three times as wide as the plate is thick 
he error resulting from the application of the usual theory exceeds 10 
>er cent of the true value of k b . It is also noteworthy that for vanishing 
Lole diameter the limit value k h = 3 of the stress-concentration factor in 
mre bending becomes equal to the value of the same factor in plane 
tress when uniform tension in one direction is assumed. 

If the hole (Fig. 165) is filled up with an elastic material other than that of the plate, 
ve have to deal with an " elastic inclusion. " The unfilled hole and the rigid inclusion 

4.0 
3.634 

3.0 



2.0 

1.0 
0.63 







v = 0.3 for material of 
plate and inclusion 




















. . * 


max/M« 






(M t 


3 
















/ y 








^l&lyox/M^ 





5 10 

Fig. 167 



15 



20 



can be regarded as the limiting cases of the elastic inclusion, Young's modulus of the 
filling being zero in the former and infinitely large in the latter case. In the following, 
the effect of a rigid inclusion is briefly considered. 

Just as in the case of a hole, we have to combine an initial state of stress with a 
supplementary one; however, the conditions now to be fulfilled on the periphery of the 
circle r = a are (in the symmetrical case) 



(W) r =a = 



dr )r-a 







(*) 



where w is the combined deflection of the plate. From the expressions (192) for the 
stress resultants, we readily conclude that on the periphery of the inclusion the relation 
Mt = vM T must hold, whereas the moments M T t become zero. 

In the particular case of pure bending, assumed on page 43, we obtain a distribu- 
tion of radial moments around the rigid inclusion given by 1 

1 M. Ooland, /. Appl. Mechanics, vol. 10, p. A-69, 1943; Fig. 167 is taken from this 
paper. See also Yi-Yuan Yu, Proc. Second U.S. Natl. Congr. Appl. Mech., Ann 
Arbor, Mich., 1954, p. 381. 



324 THEORY OF PLATES AND SHELLS 

(0 



\l + v 1 — V J 



The respective stress concentration factor is equal to k = (3 + v)/(l — ?*), that is, to 
3.63 for steel. The effect of the transverse shear deformation, however, is not implied 
in this result, which consequently holds only for large values of a/h. 

It is seen that in the vicinity of a rigid inclusion the radial couples M r far exceed 
the tangential couples Mr, this is in strong contrast to the stress state around a hole, 
where the couples M t dominate the couples M r . Both moments are balanced best 
in their magnitude in the case of an elastic inclusion, as shown in Fig. 167. Here E\ 
denotes Young's modulus of the plate and 2? 2 that of the filling. 

An inclusion with elastic filling may be replaced, without substantially changing its 
effect on the plate, by an annular elastic inclusion. Reinforcing a hole with a ring of 
properly chosen stiffness can therefore considerably reduce the stress concentration 
in the material of the plate around the hole. 1 

1 For stress analysis and numerical data regarding this case see Savin, op. cit. 



CHAPTER 10 

SPECIAL AND APPROXIMATE METHODS IN THEORY 

OF PLATES 



75. Singularities in Bending of Plates. The state of stress in a plate 
s said to have a singularity at a point 1 (xo,yo) if any of the stress com- 
ponents at that point becomes infinitely large. From expressions (101), 
^102), and (108) for moments and shearing forces we see that a singu- 
arity does not occur as long as the deflection w(x,y) and its derivatives 
xp to the order four are continuous functions of x and y. 

Singularities usually occur at points of application of concentrated 
forces and couples. In certain cases a singularity due to reactive forces 
can occur at a corner of a plate, irrespective of the distribution of the 
surface loading. 

In the following discussion, let us take the origin of the coordinates 
at the point of the plate where the singularity occurs. The expressions 
for the deflection given below yield (after appropriate differentiations) 
stresses which are large in comparison with the stresses resulting from 
loading applied elsewhere or from edge forces, provided x and y are small. 

Single Force at an Interior Point of a Plate. If the distance of the 
point under consideration from the boundary and from other concen- 
trated loads is sufficiently large, we have approximately a state of axial 
symmetry around the single load P. Consequently, the radial shearing 
force at distance r from the load P is 

Observing the expression (193) for Q r we can readily verify that the 
respective deflection is given by 

w ° = §£5 r,1 < (206) 

in which a is an arbitrary length. The corresponding term r 2 log a yields 
negligible stresses when the ratio r/a remains small. 

Single Couple at an Interior Point of a Plate, Let us apply a single 

1 More exactly, at a point (20,2/0,2). 

325 



326 



THEORY OF PLATES AND SHELLS 



force — Mi/ Ax at the origin and a single force +Mi/Ax at the point 
( — Aa;,0), assuming that Mi is a known couple. From the previous result 
[Eq. (206)] the deflection due to the combined action of both forces is 

w = Mi_ (x + Ax) 2 + y 2 lQ|r [(s + A*) 2 + */ 2 ]* 



8tD 



Ax 



a 



Mi x* + y 2 (x 2 + y 2 )* 
c - n Ax g - W 



8ttD 



a 



As Ax approaches zero, we obtain the case of a couple Mi concentrated 
at the origin (Fig. 168a) and the deflection is 



Wi = lim [w]* x -+o = 



~P"dx 



where Wo is the deflection given by expression (206). Performing the 
differentiation we obtain 



Mix/. s 2 + y 2 , A 
^ 1 = 8^V log -^- + 1 J 



W 



If we omit the second term M ix/8wD, which gives no stresses, and use 
polar coordinates, this expression becomes 



■Mi 1 r 

Wi = -t— -~ r log - cos 5 

47rZ> a 



(207) 



In the case of the couple M 2 shown in Fig. 1686 we have only to replace 
by + ?r/2 in the previous formula to obtain the corresponding 
deflection. 





Fig. 168 

Double Couple at an Interior Point of a Plate. Next we consider the 
combined action of two equal and opposite couples acting in two parallel 
planes Ax apart, as shown in Fig. 169. Putting Mi Ax = Hi and fixing 
the value of Hi we proceed in essentially the same manner as before and 
arrive at the deflection 

Hi dwi __ Hi d 2 w 

Mi dx ~~ P dx 2 



Wz 



(c) 



SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 327 



lue to a singularity of a higher order than that corresponding to a couple. 1 
Substitution of expression (206), where rectangular coordinates may be 
ised temporarily, yields the deflection 



w 2 



8tD V 



2 log - + 2 + cos 20 



a 



) 



(208) 



Expressions containing a singularity are also obtainable in the case of a 
couple acting at the corner of a wedge-shaped plate with both edges free, 
as well as in the case of a semi-infinite plate sub- 
mitted to the action of a transverse force or a 
couple at some point along the free edge. 2 

Single Load Acting in the Vicinity of a Built-in 





Fig. 169 



Fig. 170 



Edge (Fig. 170). The deflection of a semi-infinite cantilever plate carry- 
ing a single load P at some point (£,77) is given by the expression 



w 



= 1^3 H " r2 log " 



x + £) 2 + (y- v) 2 ' 



(d) 



where r\ = (.r — £) 2 -f- (y — rj) 2 . We confine ourselves to the consider- 
ation of the clamping moment at the origin. Due differentiation of 
expression (d) yields 

P 



M\ 



— COS 2 <p 
7T 



(209) 



at x = y = 0, provided £ and 77 do not vanish simultaneously. It is seen 
that in general the clamping moment M x depends only on the ratio tj/£. 

1 To make the nature of such a loading clear, let us assume a simply supported 
beam of a span L and a rigidity EI with a rectangular moment diagram Ax by M , sym- 
metrical to the center of beam and due to two couples M applied at a distance At from 
each other. Proceeding as before, i.e., making Ax —+ 0, however fixing the value of 
H = M Ax, we would arrive at a diagram of magnitude H concentrated at the middle 
of beam. Introducing a fictitious central load H/EI and using Mohr's method, we 
would also obtain a triangular deflection diagram of the beam with a maximum ordi- 
nate HL/AEI. A similar deflection diagram would result from a load applied at the 
center of a perfectly flexible string. 

2 See A. Nddai, "Elastische Platten," p. 203, Berlin, 1925. 



328 THEORY OF PLATES AND SHELLS 

If, however, £ = v\ = the moment M x vanishes, and thus the function 
M x (t,n) proves to be discontinuous at the origin. 

Of similar character is the action of a single load near any edge rigidly 
or elastically clamped, no matter how the plate may be supported 
elsewhere. This leads also to the characteristic shape of influence sur- 
faces plotted for moments on the boundary of plates clamped or continu- 
ous along that boundary (see Figs. 171 and 173). 

For the shearing, or reactive, force acting at x = y = in Fig. 170 
we obtain in similar manner 

OP 

Q z = — cos 3 <p (210) 

where r 2 = £ 2 + rj 2 . 

76. The Use of Influence Surfaces in the Design of Plates. In Art. 29 
we considered an influence function K(x,y,%,r}) giving the deflection at 
some point (x,y) when a unit load is applied at a point (£,77) of a simply 
supported rectangular plate. Similar functions may be constructed for 
any other boundary conditions and for plates of any shape. We may 
also represent the influence surface K(^rj) for the deflection at some fixed 
point (x,y) graphically by means of contour lines. By applying the 
principle of superposition to a group of n single loads P t acting at points 
(&>ifr) we find the total deflection at (x,y) as 



w 



n 

^ PiK{x,y£i,rn) (a) 



In a similar manner, a load of intensity p(^rj) distributed over an area A 
of the surface of the plate gives the deflection 

w = [[ p(t,v)K(z,y,£,ri) d$ d v (6) 

A 

By Maxwell's reciprocal law we also have the symmetry relation 

K(*,V,i,v) = K(£,ii,x,y) (c) 

i.e., the influence surface for the deflection at some point (x,y) may be 
obtained as the deflection surface w(£,rj) due to a unit load acting at 
(x,y). The surface w(£,ri) is given therefore by the differential equation 
AAw(£,ri) = 0, and the solution of this equation not only must fulfill the 
boundary conditions but also must contain a singularity of the kind 
represented in Eq. (206) at £ = x, t\ = y. 

Of special practical interest are the influence surfaces for stress resultants 1 given by 
a combination of partial derivatives of w(x,y) with respect to x and y. To take an 

1 Such surfaces have been used first by H. M. Westergaard, Public Roads, vol. 11, 
1930. See also F. M. Baron, /. AppL Mechanics, vol. 8, p. A-3, 1941 



SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 329 
example, let us consider the influence surfaces for the quantity 

d 2 w d 2 

-°* --» &*<*'•*»■> « 

By result (c) of Art. 75 this latter expression yields the ordinates of a deflection surface 
in coordinates £, y containing at £ = x, y = y a singularity due to a " couple of second 
order" H = 1 which acts at that point in accordance with Fig. 169. 

The procedure of the construction and the use of influence surfaces may be illus- 
trated by the following examples. 1 

Influence Surf ace for the Edge Moment of a Clamped Circular Plate 2 (Fig. 171). By 
representing the deflection (197), page 293, in the form w = PK(x,0,Z,0), we can con- 
sider K as the influence function for the deflection at some point (x,0), the momentary 
position of the unit load being (£,0). In calculating the edge couple M r at x = r/a = 1, 
y = we observe that all terms of the respective expressions (192), except for the 
following one, vanish along the clamped edge x = 1. The only remaining term yields 



r « _ R ( d2K \ « _ J. i 

a 2 \dx 2 ) x ^i 4tt£ 2 - 



~ l — i - (1 - f 2 ) 2 , , 



For brevity let us put £ 2 — 2£ cos + 1 = ^ 2 and, furthermore, introduce the angle 
<p (Fig. 171a). Then we have £ 2 = 1 — 2y cos <p + y 2 and 

M r = (2 cos *> — y) 2 

4ltt 

which, for negligible values of y, coincides with the expression (209). The influence 
surface for the moment M r is represented by the contour map in Fig. 1716, with the 
ordinates multiplied by 4x. 

Influence Surface for the Bending Moment M x at the Center of a Simply Supported 
Square Plate* It is convenient to use the influence surfaces for the quantities 
M x o = —2) d 2 w/dx 2 and M y o = — D d 2 w/dy 2 with the purpose of obtaining the final 
result by means of Eqs. (101). 

The influence surface for M x o may be constructed on the base of Fig. 76. The 
influence of the single load P = 1 acting at point is given by the first of the equations 
(151) and by Eq. (152). This latter expression also contains the required singularity 
of the type given by Eq. (206), located at the point 0. The effect of other loads may 
be calculated by means of the first of the equations (149), the series being rapidly 
convergent. The influence surface is shown in Fig. 172 with ordinates multiplied 
by 8tt. 

Let us calculate the bending moment M x for two single loads Pi and P 2 < Pi at a 
fixed distance of 0.25a from each other, each load being distributed uniformly over 

1 For details of the so-called singularity method see A. Pucher, Ingr.-Arch., vol. 12, 
p. 76, 1941. 

2 Several influence surfaces for the clamped circular plate are given by M. El- 
Hashimy, "Ansgewahlte Plattenprobleme, ,, Zurich, 1956. 

3 The most extensive set of influence surfaces for rectangular plates with various 
edge conditions is due to A. Pucher, "Einflussf elder elastischer Flatten," 2d ed., 
Vienna, 1958. See also his paper in "Federhofer-Girkmann-Festschrift," p. 303, 
Vienna, 1950. For influence surfaces of continuous plates, see G. Hoeland, Ingr.-Arch., 
vol. 24, p, 124, 1956. 



330 



THEORY OP PLATES AND SHELLS 




— X 




Multiplfcotion factor ^- = 0.0796 



Fig. 171 

an area 0.1a • 0.1c. Outside those areas the plate may carry a uniformly distributed 
live load of an intensity q < P 2 /0.01a 2 . 

The influence surface (Fig. 172) holds for M x0 , and the distribution of the loading 
which yields the largest value of M x0 is given in this figure by full lines. Because of the 
singularity, the ordinates of the surface are infinitely large at the center of the plate- 



SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 331 

therefore it is simplest to calculate the effect of the load Pi separately, by means of 
Eqs. (163) and (165), in connection with Tables 26 and 27. 1 For this case we have 
v = 0, v/u = k = 1, <p = 1.5708, ^ = 0, X = 2.669, and /i = 0, which yields N *= 
and a value of M calculated hereafter. As for the effect of the load P 2 , it can be 
assumed as proportional to the ordinate 2.30 of the surface at the center of the loaded 




Multiplication foctor — = 0.0398 

o7T 



For uniform lood M x =0.0369 qa 2 



Fig. 172 



area. Introducing only the excesses of both single loads over the respective loads due 
to g, wc have to sum up the following contributions to the value of M x0 : 
1. Load Pu from Eqs. (163), (165), with f = a/2, d = 0.1 V 7 2 a, 



M Pi - O.Ol^a 2 

M xQ = — = — — 

x0 2 8tt 

= 0.219(Pi - O.Olga 2 ) 



( 21og ai; 



V2 



= + 2.669 - 1.571 



) 



1 The effect of the central load may also be calculated by means of influence lines 
similar to those used in the next example or by means of Table 20. 



332 THEORY OF PLATES AND SHELLS 

2. Load P 2 : 

M" - — 2.30(P 2 - 0.01?a 2 ) - 0.092(P 2 - O.Olga 2 ) 

3. Uniform load q: from data on Fig. 172, 

M' x 'o' - 0.0369ga 2 
Therefore M x0 = 0.219Pi + 0.092P 2 + 0.0338ga 2 

Owing to the square shape of the plate and the symmetry of the boundary conditions 
we are in a position to use the same influence surface to evaluate M v o. The location 
of the load P 2 corresponding to the location previously assumed for the surface M z o 
is given by dashed lines, and the contribution of the load P 2 now becomes equal to 
My = 0.035(P 2 — O.Olqa 2 ), while the contributions of Pi and q remain the same as 
before. This yields 

M v o = 0.219Pi + 0.035P 2 + 0.0344$a 2 

Now assuming, for example, v = 0.2 we have the final result 

M x - M x0 + 0.2M v o - 0.263Pi + 0.099P 2 + 0.0407ga 2 

Influence Surface for the Moment M x at the Center of Support between Two Interior 
Square Panels of a Plate Continuous in the Direction x and Simply Supported at y = ±6/2. 
This case is encountered in the design of bridge slabs supported by many floor beams 
and two main girders. Provided the deflection and the torsional rigidity of all sup- 
porting beams are negligible, we obtain the influence surface shown 1 in Fig. 173. 

In the case of a highway bridge each wheel load is distributed uniformly over some 
rectangular area u by v. For loads moving along the center line y = of the slab a 
set of five influence lines (valid for v/b = 0.05 to 0.40) are plotted in the figure and their 
largest ordinates are given, which allows us to determine without difficulty the govern- 
ing position of the loading. Both the surface and the lines are plotted with ordinates 
multiplied by Sir. 

example of evaluation. Let us assume a = b = 24 ft in.; furthermore, for the 
rear tire P r = 16,000 lb, u - 18 in., v = 30 in., and for the front tire P/ = 4,000 lb, 
u = 18 in., v = 15 in. The influence of the pavement and the slab thickness on the 
distribution of the single loads may be included in the values u and v assumed above. 

For the rear tire we have v/b « 0.10 and for the front tire v/b « 0.05. Assuming 
the position of the rear tires to be given successively by the abscissas £ = 0.20a, 0.25a, 
0.30a, 0.35a, and 0.40a, the respective position of the front tires is also fixed by the 
wheel base of 14 ft = 0.583a. The evaluation of the influence surface for each par- 
ticular location of the loading gives a succession of values of the moment plotted in 
Fig. 173 versus the respective values of £ by a dashed line. The curve proves to have 
a maximum at about £ = 0.30a. The procedure of evaluation may be shown for this 
latter position only. 

The influence lines marked 0.10 and 0.05, respectively, yield the contribution of both 
central loads Cat y = 0) equal to 

-(16,000 • 3.24 + 4,000 • 3.32) = -65,100 lb 

and the influence surface gives the contribution of the remaining six loads as 

-16,000(1.66 + 2.25 + 0.44) - 4,000(1.59 + 2.25 + 0.41) - -86,600 lb 

1 For methods of its construction see references given in Art. 52. 




333 



334 



THEORY OF PLATES AND SHELLS 



Finally, taking into account the prescribed multiplier of l/&r = 0.0398, we have 
the result 

(M x ) mi „ = -0.0398(65,100 + 86,600) - -6,040 lb-ft per ft 

Maximum Shearing Force Due to a Load Uniformly Distributed over the Area of a 
Rectangle. A load of this type, placed side by side with the built-in edge of an infinite 
cantilever plate, is shown in Fig. 170 by dashed lines. This problem is encountered 
also in the design of bridge slabs. By using the result (210) and the principle of 
superposition we obtain the following shearing force at a; = y = 0: 



«-). 



which gives 



with 



2P f u [ v / 2 

= / dl / 

* uv JO J -i>/2 



>/2 U 2 + V 2 ) 2 



drj 



(Qx)max = « 



1 f V , Aw 2 \ V I 

a = - — log I — - + 1 J + 2 arctan — 
t \_u \v 2 J 2u\ 



(/) 



(g) 



Numerical values of the factor a are given in Table 78. As the influence of the other 
tire loads on Q x is usually negligible we have no need of an influence surface for Q x . 
The result (/) can be used with sufficient accuracy for slabs having finite dimensions 
and also, as a largest possible value, for an edge built in elastically. 

Table 78. Values of the Factor a in Eq. (/) 



v/u 


a 


v/u 


a 


0.1 


0.223 


1.2 


0.852 


0.2 


0.357 


1.4 


0.884 


0.3 


0.459 


1.6 


0.909 


0.4 


0.541 


1.8 


0.927 


0.5 


0.607 


2.0 


0.941 


0.6 


0.662 


2.5 


0.964 


0.7 


0.708 


3 


0.977 


0.8 


0.747 


4 


0.989 


0.9 


0.780 


5 


0.994 


1.0 


0.807 


10 


0.999 



77. Influence Functions and Characteristic Functions. It is interesting to note the 
close connection between the influence function (or Green's function) of the bent 
plate and the problem of its free lateral vibrations. The latter are governed by the 
differential equation 

(a) 

D dt* w 



/ d 2 d 2 \ 2 



where W(x,y,t) is the deflection, p the mass of the plate per unit area, and t the time. 
With the assumption W = w(x,y) cos pt we obtain for the function w the differential 
equation 

DAAw - \w = (b) 



SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 335 

ill which X = pV- For some specific boundary conditions, solutions of Eq. (6) 
exist only for a definite set of values Xi, X 2 , . . . , X*, . . . of the parameter X, the 
so-called characteristic numbers (or eigenvalues) of the problem. The respective solu- 
tions form a set of characteristic functions Wi(x,y), wz(x,y), . . . , w k (x,y), .... 
These functions are mutually orthogonal; i.e., 



// 



Wi(x,y)w k (x,y) dx dy = (c) 



for i ?± fc, the integral being extended over the surface of the plate. As the functions 
Wk(x,y) are defined except for a constant factor, we can "normalize" them by choosing 
this factor such as to satisfy the condition 



// 



wl(x y y) dx dy = a 2 b 2 (d) 



The form chosen for the right-hand side of (d) is appropriate in the case of a rectangular 
plate with the sides a and 6, but whatever the contour of the plate may be, the dimen- 
sion of a length must be secured for w k . The set of numbers X* and the corresponding 
set of normalized functions w k (x,y) being established, it can be shown 1 that the 
expansion 

00 

K(*,y,*,n) = -^ £ x k (e) 

&=1 

holds for the influence function of the plate with boundary conditions satisfied by the 
characteristic functions. 

By applying Eqs. (a) and (b) of the previous article to the result (e) we conclude 
that, no matter what the distribution of the loading may be, the deflection of the plate 
can always be represented by a linear combination of its characteristic functions. 

As an example, let us take the rectangular plate with simply supported edges (Fig. 
59). Eigenf unctions which satisfy Eq. (6) along with the boundary conditions 
w = Aw = and the condition (d) are 

— . mwx . niry ,~ 

ab sin sin — — \J) 

a b 

m and n being two arbitrary integers. The respective eigenvalue, from Eq. (6), is 

Substitution of this in the expansion (e) immediately leads to the result (134). For 
rectangular plates with only two opposite edges supported, the conditions on the other 
edges being arbitrary, influence functions may be obtained in a similar manner. 
However, in such a case a preliminary computation of the values of X* from the respec^ 
tive transcendental frequency equation becomes necessary. A further example of an 
influence function obtainable in the form of an expansion is the case of a circular plate, 

1 See, for instance, R. Courant and D. Hilbert, "Methods of Mathematical Physics/' 
vol. 1, p. 370, New York, 1953. 



336 



THEORY OF PLATES AND SHELLS 



for which the modes of vibration, expressible in terms of Bessel functions, are well 
known. 

78. The Use of Infinite Integrals and Transforms. Another method of treating the 
problems of bending of plates is the use of various transforms. 1 A few such transforms 
will be discussed in this article. 

Fourier Integrals. In the case of infinite or semi-infinite strips with arbitrary condi- 
tions on the two parallel edges the method of M. L6vy, described on page 113, can be 
used, but in doing so the Fourier series necessarily must be replaced by the respective 
infinite integrals. In addition to the example considered in Art. 50, the problem of an 
infinite cantilever plate (Fig. 174) carrying a single load P may be solved in this way. 2 




Fig. 174 

Let wi be the deflection of the portion AB and w 2 the deflection of the portion BC 
of the plate of width AC = a. Then we have to satisfy the boundary conditions 



wi = =0 

dx 



on x = 



d 2 W 2 d 2 W 2 



dx 2 ' ' dy 2 dx z 

together with the conditions of continuity 



aH * + (2-,)^-0 



(a) 



dxdy 2 



on x = a 



W\ = V)2 



dWi 

~dx 



dW2 

~dx 



Aw i = Aw 2 



on x = £ 



(6) 



The single force P may be distributed uniformly over a length v. Now, any even 
function of y can be represented by the Fourier integral 



2 f 00 f° 

f(y) = - / COS ay da I 

* Jo Jo 



f(rj) COS arj drj 



(C) 



Since the intensity of the loading is given by f(rj) « P/v for — v/2 < rj < v/2 and by 

1 For their theory and application see I. N. Sneddon, "Fourier Transforms," New 
York, 1951. 

2 The solution and numerical results hereafter given are due to T. J. Jaramillo, 
J. Appl. Mechanics, vol. 17, p. 67, 1950. Making use of the Fourier transform, 
H. Jung treated several problems of this kind; see Math. Nachr., vol. 6, p. 343, 1952, 



SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 337 
zero elsewhere, we have 



(d) 



On the other hand, the function f(y) is equal to the difference of the shearing forces Qz 
at both sides of the section x = £. Thus, by Eqs. (108), we have 




D — (AW! - Aw 2 ) = f(y) 
dx 



(e) 



on x = £. In accordance with Eq. (d) we represent the deflections w\ and w 2 by the 
integrals 

Wi ~ I Xi(x,a) COS ay da l = 1, 2 (/) 

in which the function 

Xi(x,a) == (-A* + -Bi^) cosh ax + (d + Dix) sinh ax 

is of the same form as the function Y m on page 114. 

It remains now to substitute expressions (/) into Eqs. (a), (6), and (e) in order to 
determine the coefficients -Ai, J?i, . . . , D 2) independent of y but depending on a. 




The distribution of bending moments along the built-in edge, as computed from the 
foregoing solution for various positions of the single load and for v = 0, v = 0.3, is 
shown in Fig. 175. 

Mellin Transform. The application of this transform is suitable in the case of a 
wedge-shaped plate with any homogeneous conditions along the edges 6 = and 



338 



THEORY OF PLATES AND SHELLS 



= a (Fig. 176). To take an example let us consider the edge = as clamped and 
the edge = a, except for a single load P at r = r , as free. 1 

We use polar coordinates (see Art. 62) and begin by taking the general solution of 
the differential equation AAw = in the form 

W(s) = r-*9(0,s) (g) 

where s is a parameter and 

0(0,s) = A(s) cos sd + B(s) sin sO + C(s) cos (s + 2)6 + D(s) sin (s + 2)0 (h) 

The deflection and the slope along the clamped edge vanish if 




The bending moment M t on the free edge 
vanishes on the condition that 

i 



d*W(s) 1 dW(s) 

v 1 

dr* r dr 

r 2 3tf 2 



J«_a 



U) 



Now, a function /(r) can be represented by means of Mellin's formula as follows: 



/(r) 



I r<r+«>i /"» 

-TH / r-'ds p'-yirfdp 

Ztn Ja-»i JO 



(k) 



where a is a real constant, subject to some limiting conditions. Specifically for a force 
P concentrated at r = ro we obtain 



f(r) = 



P [v + » i / r \ ~(«+3) 
2tITo Jff—aoi \f0/ 



w 



This suggests the following form for the deflection of the plate: 

r~ 8 e(d,s) ds 

Now, the reactive forces acting down along the edge 6 = a are given by 

(V t )e-a 



2wi J a— » % 



\ l dr J9 wma 



(I) 



(m) 



(n) 



This, by use of due expressions for Q t and M Ti (see pages 283, 284) as well as the 
expression (m), gives 



(70,. 



J) Tcr+oot 

2id y<r-oot 



— +ls* + (l -r)(« + !)(«+ 2)] 






r -(«+3) ^ ( ) 



1 The problem was discussed by S. Woinowsky-Krieger, Ingr.-Arch., vol. 20, p. 391, 
1952. Some corrections are due to W. T. Koiter, Ingr.-Arch., vol. 21, p. 381, 1953. 
For a plate with two clamped edges see Y. S. Uflyand, Doklady Akad. Nauk S.S.S.R., 
vol. 84, p. 463, 1952. See also W. T. Koiter and J. B. Alblas, Proc. Koninkl. Ned. 
Akad. Wetenschap., ser. B, vol. 57, no. 2, p. 259, 1954. 



SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 339 



We finally equate expressions (I) and (o) and thus obtain, in addition to Eqs. (i) and 
(j), a fourth condition to determine the quantities A(s), B(s) y C(s), and D(s). Sub- 
stitution of these coefficients in the expressions (h) and (m) and introduction of a new 
variable u = -(s + l)i, where i = V — 1, yields the following expression for the 
deflection of the plate: 



w = 



2Prr 



G cos 



( w log — J + # sin ( -u log — J 

Nu(l +u 2 ) ~~ ~~ 



du 



(*>) 



in which G and if are some functions of a, 6, and u } and AT is a function of a and w. 




Fig. 177 

The variation of the deflections along the free edge and the distribution of the 
moments M t along the edge = in the particular case of a = tt/4 and a = tt/2 is 
shown in Fig. 177. 

Hankel Transform. Let a circular plate with a radius a be bent to a surface of 
revolution by a symmetrically distributed load q{r). We multiply the differential 
equation AAw = g/D of such a plate by rJofar) dr and integrate by parts between 



340 THEORY OF PLATES AND SHELLS 

r = and r = <*>. Provided w = for r > a, the result is 

X 4 / w(r)rJ Q (Xr) dr = ffQO (q) 

where 

g(\) = (Oi + X*C 2 )/ (Xa) + (XC 3 + X 8 C 4 )Ji(Xa) + - / q( P )pM>») dp (r) 

Jo and Ji are Bessel functions of the order zero and one, and d are constants. Appli- 
cation of the Hankel inversion theorem to Eq. (g) gives 



w 



Too J 



Jo(Xr) dX (s) 



The constants d now are obtainable from the conditions on the boundary r — a of 
the plate and from the condition that the function g(\)/\ A must be bounded. The 
expression (r) must be slightly modified in the case of an annular plate. 1 Examples 
of the application of solutions of the type (s) to the problem of elastically supported 
plates are given in Art. 61. 

Sine Transform. In the case of rectangular plates we have used solutions of the 
form 

w(x,y) = 27(2/,a) sin ax 

and in the case of sectorial plates those of the form 

w(r,6) = XR(r,p) sin $6 

The finite sine transforms of the function w, taken with respect to x and 0, respec- 
tively, and introduced together with transformed derivatives of w and the trans- 
formed differential equation of the plate, then prove useful in calculating the constants 
of the functions Y and R from the given boundary conditions of the plate. 2 

79. Complex Variable Method. By taking z = x + i y and I «» x — iy for inde- 
pendent variables the differential equation (104) of the bent plate becomes 

q(z,z) (a) 



dz 2 dz* 16Z) 

Let us assume w = Wo + wi, where w\ is the general solution of the equation 

d A w 



d 2 z an 



= 



and Wo a particular solution of Eq. (a). Then we have 8 

wi - (R[z<p(z) + x (z)] Q>) 

where tp and x are functions which are analytic in the region under consideration. 
Usually the derivative ^ = dx/dz is introduced along with x- 

1 For the foundation of the method and an extensive list of transforms needed in its 
application see H. Jung, Z. angew. Math. Mech., vol. 32, p. 46, 1952. 

2 The application of the method is due to L. I. Deverall and C. J. Thome, J. Appl. 
Mechanics, vol. 18, pp. 152, 359, 1951. 

8 (R denotes the real part of the solution. This form of the solution of the bipoten- 
tial equation is due to E. Goursat, Bull. Soc. Math. France, vol. 26, p. 236, 1898. 



SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 341 

In the case of a single load P acting at z = xo + iyo the solution w may be chosen 
in the form 

P 

Wo = (Z - *o)(* - *o) lOg [(2 ~ Zo)(2 - 2o)] (c) 

16x2) 
which is substantially equivalent to expression (206). For a uniform load 



Wo = 



qzH 2 



64D 
would be a suitable solution. 

If the outer or the inner boundary of the plate is a circle we always can replace it 
by a unit circle z = e ie , or briefly z = a. The boundary conditions on z = a must be 
expressed in complex form also. The functions <p and ^ may be taken in the form of a 
power series, with additional terms, if necessary, depending on the value of stress 
resultants taken along the inner edge of the plate. Multiplication of the boundary 
conditions by the factor [2>jri(<r - z)]" 1 da and integration along z = <r then yields the 
required functions <p and \f/.* 

For boundaries other than a circle a mapping function z = w(f) = co(pe^) may be 
used so as to map the given boundary line onto the unit circle f = e % * = <r. The 
determination of the functions *>i(f) = <p(z) and *i(f) = ^(z) from the boundary 
conditions on f = c then is reduced to the problem already considered. The Musch- 
eliShvili method outlined above is especially efficient in cases concerning stress 
distribution around holes; 1 the function w(f) then has to map the infinite region of the 
plate into the interior of the unit circle. 

The complex variable method also allows us to express Green's functions of a circular 
plate with various boundary conditions in closed form. 2 In other cases, such as that 
of a clamped square plate, we must rely on an approximate determination of the 
Green functions. 3 

When expressible by a double trigonometric series, the deformation of the plate can 
also be represented in a simpler form by making use of the doubly periodic properties 
of the elliptic functions. For the quantity Aw, satisfying the potential equation 
A (Aw) = 0, such a representation becomes particularly convenient because of the 
close connection between the Green function for the expression Aw and the mapping 
function of the region of the given plate into the unit circle. 4 Once Aw is determined 

*For evaluation of integrals of the Cauchy type implied in this procedure see N. I. 
Muschelighvili, "Some Basic Problems of the Mathematical Theory of Elasticity," 
Groningen, 1953. 

1 An extensive application of the method to the problem of stress concentration is 
due to G. N. Savin; see his " Stress Concentration around Holes, ,, Moscow, 1951. 
See also Yi-Yuan Yu, J. Appl. Mechanics, vol. 21, p. 129, 1954, and Proc. Ninth 
Intern. Congr. Appl. Mech., vol. 6, p. 378, Brussels, 1957; also L. I. Deverall, J. Appl. 
Mechanics, vol. 24, p. 295, 1957. A somewhat different method, applicable as well 
to certain problems of the thick-plate theory, was used by A. C. Stevenson, Phil. Mag., 
vol. 33, p. 639, 1942. 

2 E. Reissner, Math. Ann., vol. Ill, p. 777, 1935; A.Lourye, Priklad. Mat. Mekhan., 
vol. 4, p. 93, 1940. 

3 F. Schultz-Grunow, Z. angew. Math. Mech., vol. 33, p. 227, 1953. 

4 Gourant and Hilbert, op. cit., vol. 1, p. 377. Elliptic functions have been used in 
particular by A. Nddai, Z. angew. Math. Mech., vol. 2, p. 1, 1922 (flat slabs); by 
F. Tolke, Ingr.-Arch., vol. 5, p. 187, 1934 (rectangular plates); and also by B. D. 
Aggarwala, Z. angew. Math. Mech., vol. 34, p. 226, 1954 (polygonal plates and, in 
particular, triangular plates). 



342 THEORY OF PLATES AND SHELLS 

the shearing forces of the plate are readily given by the derivatives of that function by 
virtue of Eqs. (108). 

80. Application of the Strain Energy Method in Calculating Deflec- 
tions. Let us consider again the problem of the simply supported 
rectangular plate. From the discussion in Art. 28 it is seen that the 
deflection of such a plate (Fig. 59) can always be represented in the form 
of a double trigonometric series: 1 

80 00 

h h sm ~r sm ~r (a) 

m=l n« 1 

The coefficients a mn may be considered as the coordinates defining the 
shape of the deflection surface, and for their determination the principle 
of virtual displacements may be used. In the application of this principle 
we need the expression for strain energy (see page 88) : 

Substituting series (a) for w, the first term under the integral sign in (b) 
becomes 



oo oo 



l * D L Jo [^L / am "(^ + ^) sin ^ sin ^J dxd y (*) 



Observing that 



f a . rmrx . m'-irx , f b . nicy . u'-kxi , 

/ sm — — sm — — dx = / sm -J2 s i n 'III dy = 
Jo a a Jo b b ■ 

if m ?* m' and n ^ n', we conclude that in calculating the integral (c) 
we have to consider only the squares of terms of the infinite series in the 
parentheses. Using the formula 

f a f b . rmrx . 9 mry , . ab 

/ / sin 2 sin 2 -^ dx dy = — 

Jo Jo a b * 4 

the calculation of the integral (c) gives 

» 2 £<■&+# 



00 00 

T 

m—l n=*l 



1 The terms of this series are characteristic functions of the plate under consideration 
(see Art. 77). 



SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 343 

From the fact that 



a 6^4 



jl SI™' ^ ** ? * * " II II cos ' =¥ cos! T * d " - T 

it can be concluded that the second term under the integral sign in 
expression (6) is zero after integration. Hence the total strain energy 
in this case is given by expression (c) and is 

m=l n=l 

Let us consider the deflection of the plate (Fig. 59) by a concentrated 
force P perpendicular to the plate and applied at a point x = £, y = v . 
To get a virtual displacement satisfying boundary conditions we give to 
any coefficient a m v of series (a) an infinitely small variation 8a m ' n '. As a 
result of this the deflection (a) undergoes a variation 

5w = 5a m v sin sin -t-£ 

a 6 

and the concentrated load P produces a virtual work 

" oa m 'n' sin sin — ^ 

From the principle of virtual displacements it follows that this work must 
be equal to the change in potential energy (d) due to the variation 5a m ' n '. 
Hence 

P 8a m > n > sin — -* sm -=--* = Sa m v 

a o oa m ' n ' 

Substituting expression (d) for V, we obtain 

d *„ • w '*"£ • nVif T 4 ab ~ /ra' 2 , n /2 \ 2 

P 5a wV sm __ sin _^ = _. Da^v (^ + ^r) Sa mV («) 

from which 

4 p s in^sin^p 

Substituting this into expression (a), we obtain once more the result (133). 
Instead of using the principle of virtual displacements in calculating 
coefficients a mn in expression (a) for the deflection, we can obtain the 
same result from the consideration of the total energy of the system. 
If a system is in a position of stable equilibrium, its total energy is a 



344 THEORY OF PLATES AND SHELLS 

minimum. Applying this statement to the investigation of bending of 
plates, we observe that the total energy in such cases consists of two 
parts: the strain energy of bending, given by expression (6), and the 
potential energy of the load distributed over the plate. Defining the 
position of the element q dx dy of the load by its vertical distance w from 
the horizontal plane xy, the corresponding potential energy may be taken 
equal to — wq dx dy, and the potential energy of the total load is 

- / Jwq dx dy (g) 

The total energy of the system then is 

The problem of bending of a plate reduces in each particular case to 
that of finding a function w of x and y that satisfies the given boundary 
conditions and makes the integral (h) a minimum. If we proceed with 
this problem by the use of the calculus of variations, we obtain for w 
the partial differential equation (104), which was derived before from the 
consideration of the equilibrium of an element of the plate. The inte- 
gral (h), however, can be used advantageously in an approximate investi- 
gation of bending of plates. For that purpose we replace the problem of 
variational calculus with that of finding the minimum of a certain func- 
tion by assuming that the deflection w can be represented in the form 
of a series 

w = aKpi(x,y) + a2<p 2 (x,y) + a z <p z (x,y) + • • • + a n <p n (x,y) (211) 

in which the functions <pi, <p%, . . . , <p n are chosen so as to be suitable 1 
for representation of the deflection surface w and at the same time to 
satisfy the boundary conditions. Substituting expression (211) in the 
integral (h)> we obtain, after integration, a function of second degree with 
coefficients ai, a^ . . . . These coefficients must now be chosen so as 
to make the integral (h) a minimum, from which it follows that 

dI n dI n dI n ^ 

— - = t— = ••• — = (i) 

dai da,2 da n 

This is a system of n linear equations in ai, a*, . . . , a n , and these 
quantities can readily be calculated in each particular case. If the 
functions <p are of such a kind that series (211) can represent any arbi- 

1 From experience we usually know approximately the shape of the deflection 
surface, and we should be guided by this information in choosing suitable functions <p. 



SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 345 

trary function within the boundary of the plate, 1 this method of calcu- 
lating deflections w brings us to a closer and closer approximation as the 
number n of the terms of the series increases, and by taking n infinitely 
large we obtain an exact solution of the problem. 

Applying the method to the case of a simply supported rectangular 
plate, we take the deflection in the form of the trigonometric series (a). 
Then, by using expression (d) for the strain energy, the integral (h) is 
represented in the following form: 



00 00 



_ Tc'abD v V 2 (™1 4. *IY 

8~ 2/ Z, ttmn \a> + V) 

- / a I q V V a mn sin ^^ sin ^ dx dy (j) 



7W= 1 n = 1 

oo oo 



m=l n=l 



and Eqs. (i) have the form 

Tr A abD /m 2 , n 2 \ 2 f a f h . rmrx . nicy , , A /7N 

In the case of a load P applied at a point with the coordinates £, ?/, 
the intensity 5 of the load is zero at all points except the point £, 77, where 
we have to put qdxdy = P. Then Eq. (fc) coincides with Eq. (e), previ- 
ously derived by the use of the principle of virtual displacements. For 
practical purposes it should be noted that the integral 

contained in expressions (6) and (h) vanishes for a plate rigidly clamped 
on the boundary. The same simplification holds for a polygonal plate 
if one of the boundary conditions is either w = or dw/dn = 0, where 
n = direction normal to the edge. 2 

If polar coordinates instead of rectangular coordinates are used and 
axial symmetry of loading and deformation is assumed, Eq. (h) has to be 
replaced by 

1 We have seen that a double trigonometrical series (a) possesses this property with 
respect to deflections w of a simply supported rectangular plate. Hence it can be used 
for obtaining an exact solution of the problem. The method of solving the bending 
problems of plates by the use of the integral (h) was developed by W. Ritz; see J. reine 
angew. Math,, vol. 135, 1908; and Ann. Physik, ser. 4, vol. 28, p. 737, 1909. 

2 See, for instance, E. R. Berger, Osterr. Ingr.-Arch., vol. 7, p. 41, 1953. 



346 THEORY OF PLATES AND SHELLS 

The contribution of the term containing the factor 1 — v again is zero 
for a plate clamped along the boundary. 

The strain energy method can also be used for calculating the deflection of a circular 
plate resting on an elastic foundation. For example, to obtain a rough approximation 
for the case of a circular plate, we take for the deflection the expression 

w - A + Br 2 (n) 

in which A and B are two constants to be determined from the condition that the 
total energy of the system in stable equilibrium is minimum. 
The strain energy of the plate of radius a as given by Eq. (m) is 

Vi = 4£ 2 Z>7ra 2 (l + v) 

The strain energy of the deformed elastic foundation is 

V 2 = / / —rdrde-irkl- AW + - ABa* + - B*a* ) 
Jo Jo 2 \2 2 6 / 

The total energy of the system for the case of a load P applied at the center is 

V = 4B*Dira*(l +v)+ irk(iAW + ^ABa* + \BW) - PA . 

Taking the derivatives of this expression with respect to A and B and equating them 
to zero, we obtain 

16Z>(1 + v) 



A + Ba 2 - + 



fca 4 



]- 



A +^5a 2 - 



2 irka 2 

In accordance with the numerical example on page 264 we take 

I = a — - 1 —;— - 102 • 10-* 
ka A 8irka z 

and obtain 

WW = A - 41.8 • 10~ 3 in. 

This result is about 3 per cent less than the result 43 • 10" 3 obtained from the differen- 
tial equation of a plate resting on elastic foundation. For greater accuracy more 
terms should be taken in expression (n). 

If the stress distribution around the single load, not merely the deflection, were 
desired, a term of the form 

r 2 log r 

should be included in expression (n) in accordance with the type of singularity here 
required [see Eq. (206)]. 

When using polar coordinates in the most general case the integral (A) assumes the 
form 

ff \D \(d*w , 1 dw 1 dhv\ 2 n . . d 2 w (l dw 1 d 2 uA 

N (\ d 2 w 1 dwy] ) , JO , N 

+ 2(1 - v) ( — J - wqi r dr d$ (o) 

^ v ' \r dr d$ r 2 86/ J l ) 



SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 347 

81. Alternative Procedure in Applying the Strain Energy Method. 

The calculation of the coefficients ai,a 2 , . . . ,a n in expression (211), which 
had to satisfy the boundary conditions but not the differential equation 
of the problem, may also be carried out without actually determining 
the potential energy of the system. 

Let us assume a virtual deflection 8w of the plate; then, we can calcu- 
late the respective work of the loading q either directly, by means of the 
integral 

(SV)i = jjq 8w dx dy (a) 

or indirectly, using the expression 

(57)2 = JfDAAw 8w dx dy (b) 

If w were the exact solution of the differential equation DAAw = q of the 
plate, then the expressions (a) and (b) would be identical. For an 
approximate solution, which Eq. (211) represents, this is certainly not 
the case. We can succeed, however, in equalizing the expressions for the 
work for a particular set of virtual deflections, namely for 8wi = <pi 8a,i, 
8w 2 = <f2 8a 2 , . . - , 8w n = ip n 8a n . Substituting these expressions con- 
secutively in the equation (8V)i = (8V) 2 or, what is the same, in the 
equation 

Jjq 8w dx dy = j J DAAw 8w dx dy (c) 

we obtain the following system of equations: 1 

ff (aAw - ^\ <p x dx dy = 
// (AAw - -^ J <p 2 dx dy = 

[J (aAw - ^\ <p n dx dy = 

It remains only to substitute the expression (211) in Eqs. (d) and to 
resolve them with respect to the unknown coefficients cti, a 2 , . . ., a n . 
This leads to the final expression for the deflection (211). 

To illustrate the application of the method let us consider a uniformly 
loaded rectangular plate with all edges built in (Fig. 91). Writing for 
brevity 2x/a = u, 2y/b = v, we shall use the expressions 



Ui = u* - 2u 2 + 1 Vi = v 4 - 2*/ 2 + 1 

U 2 = u« - 2u A + u 2 V 2 = v« - 2v A + v 2 



w 



1 The principle leading to those so-called Galerkin equations was indicated by 
W. Ritz; see "Gcsammcltc Wcrke," p. 228, 1911. 



348 THEORY OF PLATES AND SHELLS 

The set of functions 

vi = UiVt <P2 = UxV* <p z = V%Vx <P4 = UtVt (/) 
then fulfills the required conditions 

dw A _±_1 

and w = -r— = onv = ±1 

Let us carry out the computation for the particular case of the square plate. As 
x and y now are interchangeable, we have a 2 = a 3 and, consequently, 

<pi = <pz = C/1F2 + J72^i 

Putting ga 4 /16Z> = AT we take expression (211) in the form 

w - a^tFi + a,(^i7, + CW) + a 4 [/ 2 7 2 ((7) 

Substituting this consecutively in Eqs. (d) with the factors <p\, <p2, and <p* and 
observing notation (e) we have then to evaluate the integrals between the limits 
u =» ±1, v = ±1. Thus we arrive at the following system of equations: 

6.687345ai + 1.215879a 2 + 0.0675488a 4 - 0.1422221AT 

1.215879ai + 2.743525a 2 + 0.218235a 4 - 0.0406349iV (h) 

0.0675488ai + 0.218235a 2 + 0.00590462a 4 = 0.00290249JV 

For the first approximation we have 

0.1422221 ^ r „^^ XT 
a i - n.«^*e N - 0.02127# 
6.687345 

Resolving the whole system (h) we have 

ax - 0.02023i\T a 2 = 0.00535W a 4 - 0.00625# 

for the third approximation. 

Numerical results obtained by means of the expression (g) for the deflection at the 
center, the moments M x = M y at the center, and the moment M x at x = a/2, y = 0, 
respectively, are the following: 

First approx. 0.001329ga 4 /A 0.0276ga 2 , -0.0425ga 2 

Third approx. 0.001264ga 4 /Z>, 0.0228ga 2 , -0.0512ga 2 

For comparison, Table 35 gives the values 

0.00126gaVA 0.0231ga 2 , -0.0513ga* 

The moments at the center are calculated for v = 0.3. 

It is seen that, whereas the first approximation is not yet satisfactory, the third 
approximation appears quite sufficient even for the bending moments concerned. 

82. Various Approximate Methods. A Combined Method. 1 The procedure 
described in the foregoing article may be restricted as well to one variable, say y, thus 
obtaining for the other variable, x, an ordinary differential equation. Let us consider 
again the bending of a clamped square plate under uniform load (Fig. 91). 

1 Due to L. V. Kantorovich, Izvest. Akad. Nauk S.S.S.R., no. 5, 1933. 



SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 349 

In confining ourselves to the first approximation we take, this time, 

w = v(x)f(y) = <p(x)(a* - 8a V + 16t/ 4 ) (a) 

the boundary conditions w = dw/dy = on y = ±a/2 thus being fufilled by the 
function +{y). Now we try to satisfy the condition (c) of Art. 81 by choosing the 
variation in the form 

8w = *(y) 8<p(x) (*>) 

This, after substitution in Eq. (c) of Art. 81, yields 

f T f (aAw - -M HV) dy dx 8<p(x) = (c) 

which is fulfilled if 

/ ( AAw - |j +(y) dy = id) 

Next, we substitute expression (a) in this latter equation and obtain the following 
differential equation for the unknown function <p(x) : 

a 4 dV __ <^ dhp ^ q ,. 

504 ds 4 21 dx 2 * 384D 

An obvious particular solution of this equation is <p = g/384D. For the homogeneous 
equation resulting from Eq. (e), when q = 0, we have to assume <p = e x * /a . This 
yields X = ±a ± pi, with a = 4.1503 and p = 2.2858. In view of the symmetry 
of the deflection surface about the y axis, solutions of Eq. (e) must be even functions 
in x; accordingly we have 

/ „ , ax px . ax . px\ //x 

[ 1 + Ci cosh — cos h C 2 smh — sin — 1 (/) 

y a a a a/ 

To calculate the constants d and C 2 we use the boundary conditions *> = d<p/dx = 
on x = ±a/2. Thus we obtain Ci = -0.50227, C 2 = -0.04396, which establishes 
definitively the form of the function (/) and the solution (a). 

We derive from this latter the following numerical results for the center of the plate: 
w = 0.001296ga 4 /D and (for v = 0.3) M x = 0.0241ga 2 and M v » 0.0261ga 2 . 

Owing to the partial use of the differential equation the results of the first approxi- 
mation prove to be more exact than those of Art. 81, where a pure strain energy method 
was applied. To improve the accuracy still further, we have to assume 

w = <pi(x)*i(x) + <P2(x)+ 2 (f) + • • • (9) 

where all the functions tiy) have to fulfill the boundary conditions ont/= ±a/2. 
The use of Eq. (c) in conjunction with the variations 8wi = ^1 6*>i, dw 2 = ^2 &<P2, . • • 
would lead this time to a system of linear differential equations with constant coeffi- 
cients for the functions *>i(x), <p 2 (x), .... The handling of such a system, though 
simple in principle, may become troublesome for higher approximations; the second 
approximation, however, should be adequate for the most practical purposes. 

The Method of Reversion. Solution (211), fulfilling only the boundary conditions 
of the problem, may also be used in the following manner. Instead of calculating the 
deflections from a given load distribution by means of the differential equation (103) 
we use the same equation to calculate the loading 

q = DAAw (h) 



<P = 



384D 



350 



THEORY OF PLATES AND SHELLS 



resulting from the tentative expression (211) for the deflection. According to our 
hypothesis, expression (211) does not represent the rigorous solution of the problem 
and, therefore, the loading (h) will never be identical with the given loading q. We 
can, however, choose the parameters a if a 2 , . . . in Eq. (211) so as to equalize the 

functions q and q on the average over some por- 
tions of the area of the plate. 

Consider, for example, a rectangular plate 
(Fig. 178) with boundary conditions and a dis- 
tribution of loading symmetrical about both 
axes x and y. Having subdivided the plate 
into 16 equal rectangles, we need, because of 
* the symmetry, to consider only four partial 
areas, such as A\, A 2, A*, and A 4 . Expression 
(211) can be restricted accordingly to four 
terms, i.e., to 



I 



'"1 



< ¥r 



4" 



i 
i 
i 
i 

-£>|CM 

I 

I 
I 

I 



4- 



-o|sr 



* 



I 



w = a\<p\ + a 2 <p2 + anpi + a t <pi 



(0 



Now let q and q' undergo in each of the partial 
areas the condition 



Fig. 178 



// « - s) 



dx dy = n - 1, 2, 3, 4 (;) 



This gives four linear equations for the four parameters a n and the resolution of these 
equations establishes the expression (i) in its final form. 1 

Methods Approximating the Boundary Conditions. If we succeed in finding a solu- 
tion which fulfills the differential equation (103) together with one of the boundary 
conditions, the second prescribed condition may be satisfied by determination of a set 
of suitably chosen parameters. In solving the problem stated in Art. 44 coefficients 
of the two trigonometric series representing the variation of the edge moments of the 
plate were introduced as such parameters. Expansion of the slope dw/dN in Fourier 
series 2 along the boundary was used in order to let this slope vanish in accordance 
with the requirements of the problem. In using the latter condition the parameters 
could be calculated. Some minimum principle — for example, the method of least 
squares — may be used as well in order to satisfy approximately the conditions on the 
boundary. The application of such a principle needs more detailed consideration 
when two boundary conditions must be simultaneously fulfilled. 3 

In using a solution which satisfies only the differential equation of the problem it 
sometimes proves simplest to fulfill the boundary conditions merely at a number of 
points suitably chosen along the boundary. The symmetry of the deformation of the 
plate, if such a symmetry exists, should be taken into account in locating those points. 
In order to satisfy all boundary conditions at m points we must introduce 2ra unknown 
parameters. 

In the most general case 4 we may use an expression for the deflection which satisfies 
neither the differential equation of the bent plate nor the boundary conditions of the 

1 An illustrative example for the application of the method may be found in C. B. 
Biezeno and R. Grammel, "Technische Dynamik," 2d ed., vol. 1, p. 147, Berlin, 1953. 

1 A more general system of functions orthogonalized along an edge was used by 
A. N&dai to fulfill a boundary condition; see "Elastische Platten," p. 180, Berlin, 1925. 

3 An important contribution to this question is due to E. Berger, op. cit., p. 39. 

4 The method was discussed by C. J. Thorne and J. V. Atanasoff, Iowa State Coll. J. 
Sci., vol. 14, p. 333, 1940. 



SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 351 

problem. A number of points, say n, will be chosen then on and inside the boundary 
of the plate in which the differential equation must be satisfied exactly. Therefore 
a total of 2m + n parameters will be needed to obtain the solution of the problem. 

Weinstein's Method. 1 In the specific case of a plate built in along the boundary we 
may seek at first a solution of the differential equation A Aw i = q/D such that the 
solution is valid for the given loading q and for the boundary conditions w x = 0, 
& Wl = o, instead of the actual conditions. It has been shown in Art. 24 that this latter 
procedure is equivalent to solving in succession two problems, each dealing with the 
equilibrium of a loaded membrane. 

The solution of the actual problem may be taken in the form 



W — W\ + 






fc=i 



(0 



where a k are some coefficients and <p k functions of x, y, vanishing at the boundary and 
obeying the differential equation AA<p k = 0. The required condition dw/dN at the 
boundary (where N is the normal to the boundary) can be modified by means of 
Green's theorem, which leads to the following system of m linear equations for the 
parameters a k : 

m 

J — dx dy + y a k A<pi A<p l: dx dy - 
k = l 

m 

// ^ dx dy + } a k J A<p 2 A<p k dx dy = 
fc= l 

where all integrals are taken over the entire area of the plate. The method may be 
used to advantage when the boundary conditions w = 0, Aw = suggest a much 
simpler solution of the problem than the actual conditions w = 0, dw/dN — 0. 

83. Application of Finite Differences Equations to the Bending of 
Simply Supported Plates. In our previous discussion (see Art. 24) it 
was shown that the differential equation for the bending of plates can be 
replaced by two equations each of which has the form of the equation 
for the deflection of a uniformly stretched membrane. It was mentioned 
also that this latter equation can be solved with sufficient accuracy by 
replacing it by a finite differences equation. To illustrate this method of 
solution let us begin with the case of a uniformly loaded long rectangular 
plate. At a considerable distance from the short sides of the plate the 
deflection surface in this case may be considered cylindrical. Then, by 
taking the x axis parallel to the short sides of the plate, the differential 
equations (120) become 

-^r = -q 

dx (a) 

d 2 w _ M 

dx* ~ D 
'A. Weinstein and D. H. Rock, Quart. Appl. Math., vol. 2, p. 262, 1944. 



352 



THEORY OP PLATES AND SHELLS 



Both these equations have the same form as the equation for the deflec- 
tion of a stretched and laterally loaded flexible string. 

Let AB (Fig. 179) represent the deflection curve of a string stretched 
by forces S and uniformly loaded with a vertical load of intensity q. In 
deriving the equation of this curve we consider the equilibrium of an 
infinitesimal element mn. The tensile forces at points m and n have the 




n^zt 



i_ dw . d 2 w 



dx dx* 



dx 



b K-1 X VVK W 



K+1 




(b) 
Fig. 179 



directions of tangents to the deflection curve at these points; and, by 
projecting these forces and also the load q dx on the z axis, we obtain 



S 



from which 



dw ~ / dw 
dx \dx 



+ g'<te) + 9 fo = 



d 2 W _ £ 

dx 2 ~ S 



Q>) 



(c) 



This equation has the same form as Eqs. (a) derived for an infinitely 
long plate. The deflection curve is now obtained by integrating Eq. (c), 
which gives the parabolic curve 

48a; (a — x) 



w = 



a* 



(d) 



satisfying the conditions w = at the ends and having a deflection 8 at 
the middle. 



SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 353 

The same problem can be solved graphically by replacing the uniform 
load by a system of equidistant concentrated forces q Ax, Ax being the 
distance between two adjacent forces, and constructing the funicular 
polygon for these forces. If A (Fig. 179) is one of the apexes of this 
funicular polygon and S k -i and S k are the tensile forces in the two adja- 
cent sides of the polygon, the horizontal projections of these forces are 
equal to £ and the sum of their vertical projections is in equilibrium with 
the load q Ax, which gives 



W k — Wfc-i , g W k +1 — Wk 



Ax 



Ax 



+ q Ax = 



(<0 



In this equation w k -i, w k , and w k+ \ are the ordinates corresponding to the 
three consecutive apexes of the funicular polygon, and (w k — w k -i)/Ax 
and (w k+ i — w k )/Ax are the slopes of the two adjacent sides of the poly- 
gon. Equation (e) can be used in calculating the consecutive ordinates 
w h it>2, ... , w*-i, u>k, Wk+i, . . . , Wn of the funicular polygon. For 
this purpose let us construct Table (/). 






Wo 


Awo 




Ax 


W\ 






















Awk-i 
Awj, 




(k - 1) Ax 


Wk-l 




k Ax 


w k 


A 2 Wk 


(k + 1) Ax 


Wk+1 
















- 



(/) 



The abscissas of the consecutive division points of the span are entered in 
the first column of the table. In the second column are the consecutive 
ordinates of the apexes of the polygon. Forming the differences of the 
consecutive ordinates, such as Wi — w 0} . . . , w k — w k -i, w k +i — u> k , 
. . . , we obtain the so-called first differences denoted by Aw , . . . , 
Aw k -i, Aw k , . ... , which we enter in the third column of the table. The 
second differences are obtained by forming the differences between the 
consecutive numbers of the third column. For example, for the point fc 
with the abscissa k Ax the second difference is 

Aho k — Aw* — Aw k -\ 

= w k+ i — Wk — (w k - Wft-i) = Wk+i - 2w* + w*-i (g) 



354 THEORY OF PLATES AND SHELLS 

With this notation Eq. (e) can be written in the following form: 

A 2 w q 



(h) 



Az 2 S 

This is a finite differences equation which corresponds to the differential 
equation (c) and approaches it closer and closer as the number of division 
points of the span increases. 

In a similar manner the differential equations (a) can be replaced by 
the following finite differences equations: 

A 2 M 

Ax 2 9 n 

A*w ^ _M W 

Ax 2 D 

To illustrate the application of these equations in calculating the deflec- 
tions of the plate let us divide the span, say, into eight equal parts, i.e., 
let Ax = |a. Then Eqs. (i) become 

AW = - g 



A 2 w = — 



64 

Ma 2 

64D 



Forming the second differences for the consecutive division points Wi, 
w 2 , W3, and w 4 in accordance with Eq. (g) and observing that in our case 
Wo = and M = and from symmetry w z = w$ and M 3 = ilf 5 , we 
obtain the two following groups of linear equations: 



M 2 - 2M X = - g 2 


o M> 2 


M 3 - 2M* + M 1 = - |£ 


w 3 - 2w 2 + wi = 64£) 


M i - 2M 3 + M 2 = - g£ 


w 4 2w 3 + w 2 = 64£> 


M 3 - 2M t + M 3 = - |j 


w z 2wi + w 3 = — ^^ 


g the first group, we obtain the following values for M: 


»>-\t *•-«£ 


, , 15 oa 2 ,, go 2 



0) 



(*) 

These values coincide exactly with the values of the bending moments 
for a uniformly loaded strip, calculated from the known equation 



SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 355 

Substituting the values (fc) for the moments in the second group of Eqs. 

0), we obtain 

W2 — 2w\ = —%N 
w z — 2w 2 + Wi = — 6N 

w A — 2w z + w 2 = — -yw 

w 3 — 2w 4 + w-i — — 8Af 



where 



64 2 D 



Solving these equations, we obtain the following deflections at the division 
points: 

u>! = 21 AT u> 2 = 38.5 JV w 3 = 50 AT w 4 = 54 iV (I) 

The exact values of these deflections as obtained from the known equation 



w = 



qx 
241) 



(a 3 - 2a.r 2 + x 3 ) 



X 



•Ax, Axi 



Ay 



rryvfl 
"ST 



for the deflection of a uniformly loaded strip of length a, for purposes of 
comparison, are 

wi = 20.7N to* = 38JNT w 3 = 49.42V w 4 = 53.3AT 

It is seen that by dividing the span into eight parts, the error in the 
magnitude of the maximum deflection as obtained from the finite differ- 
ences equations (i) is about 1.25 per cent. By increasing the number of 
division points the accuracy of our calculations can be increased; but this 
will require more work, since the number of 
equations in the system (j) increases as we 
increase the number of divisions. 

Let us consider next a rectangular plate 
of finite length. In this case the deflections 
are functions of both x and y, and Eqs. 
(a) must be replaced by the general equa- 
tions (120). In replacing these equations 
by the finite differences equations we have 
to consider the differences corresponding to 
the changes of both the coordinates x and y. 
We shall use the following notations for the first differences at a point 
Amn with coordinates m Ax and n Ay. The notation used in designating 
adjacent points is shown in Fig. 180. 

A x W m -i, n = Wmn ~ Wm-l.n A x W mn = Wm+l.n — Wmn 

A y W mt n-l = Wmn — Wm,n-1 A y W mn = W m , n +1 — Wmn 

Having the first differences, we can form the three kinds of second differ- 
ences as follows: 



Ay » Am-I^A 



m.n 






Fig. 180 



356 



THEORY OF PLATES AND SHELLS 



A xx W mn = A x W mn — 



AyyWmn = AyWmn — 



A x tP m _i,n = W«+l,» — W»» — (W m » — W m -l,n) 

= tOm+l.n — 2w mn + «J m -l,n 
A„t0 m ,„_i = W m , n+ i — W mn — (w mn — W»,«-l) />> 

— W m>n+ i — 2w mn + Wm,n-1 
AxyWmn = A„t(W — A v tt) m _i,„ = W m ,n+1 ~ « mn — (uV-l.n+1 — Wm-l.n) 

= M>m,n+1 — Wmn — Wm-l.n+1 + «>i»-l.n 

With these notations the differential equations (120) will be replaced by 
the following differences equations: 



A XX M , AyyM 



Ax 2 

A xx w 

Ax 2 



+ 



Ay 2 



= ~Q 



■ A yv W 

"•" Ay 2 



M 
D 



(n) 




In the case of a simply supported rectangular plate, M and w are equal to 
zero at the boundary, and we can solve Eqs. (w) in succession without any 

difficulty. 

To illustrate the process of calculating 
moments and deflections let us take the very 
simple case of a uniformly loaded square 
plate (Fig. 181) . A rough approximation for 
M and w will be obtained by dividing the 
plate into 16 small squares, as shown in the 
figure, and by taking Ax = Ay = a/4 in Eqs. 
(n). It is evident from symmetry that the 
calculations need be extended over an area 
of one-eighth of the plate only, as shown in 
the figure by the shaded triangle. In this 
area we have to make the calculations only for the three points 0, 1, 2, 
for which M and w are different from zero. At the remaining points 3, 4, 5, 
these quantities are zero from the boundary conditions. Beginning with 
the first of the equations in) and considering the center of the plate, point 
0, we find the following values of the second differences for this point by 
using Eqs. (w) and the conditions of symmetry: 

A XX M o = 2Mi - 2M 

AyyM* = 2Mi - 2M 

in which Mi and M are the values of M at points 1 and 0, respectively. 
Similarly for point 1 we obtain 

A XX M X = M s - 2Mi + Mo = -2Mi + M 
AyyMi = 2Mt - 2Mi 

The second differences at point 2 can be calculated in the same way. 
Substituting these expressions for the second differences in the first of 



SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 357 

the equations (n), we obtain for points 0, 1, and 2 the following three 
equations: 

4Mi - 4M «= 



16 



2M 2 - 4Mx + Mo = 
-4M 2 + 2Mi = 



from which we find 

Mo = 



2 64 



TIT 7 ? a2 

Ml = 2li 



16 
ga 2 
16 



M 2 = ^^ 



11 £^ 2 

4 64 



Substituting these values of moments in the second of the equations (n), 
we obtain the following three equa- 
tions for calculating deflections Wo, 
Wi, and w 2 : 

4wi — 4w = — 1 N 
2w 2 — 4w x + Wo = — i-N 
-4w 2 + 2^i = -_V-^ 



where 



AT = 



ga 4 



16 • 64Z) 



From these equations we find the fol- 
lowing values of the deflections : 

wo = &N w 1 = ff AT w 2 = ff tf 

For the deflection at the center we 
obtain 





\ 




-> 


Ax 


^ 

<— 


\ 




1 




\ 




1 


2 


4 


7 


\ 


X 








3 


5 


8 


5 


X ? 


«- 








6 


7 


4 


2 


\ 




\ 


*-Z€?^ 


\ 












\ 






< 


2 
"3 


b- 


— > 


<-- 




— 1 


) — 





-4 



Fig. 182 



66 Af 



66ga 4 



16 • 64/) 



0.00403 ^j 



Comparing this with the value 0.00406ga 4 /Z) given in Table 8, it can be 
concluded that the error of the calculated maximum deflection is less than 
1 per cent. For the bending moment at the center of the plate we find 

which is less than the exact value 0.0479ga 2 by about 4£ per cent. It 
can be seen that in this case a small number of subdivisions of the plate 
gives an accuracy sufficient for practical applications. By taking twice 
the number of subdivisions, i.e., by making Ax = Ay = £a, the value 
of the bending moment will differ from the exact value by less than 
1 per cent. 

As a second problem let us consider the bending of a simply supported skew plate 
carrying a uniform load of intensity q (Fig. 182). The subdivisions in this case are 



358 THEORY OP PLATES AND SHELLS 

Ax = 6/6 and Ay = 6/3. Therefore the first of the equations (n) can be written as 

qb 2 
4A XX M + A vy M - - Y (o) 

Applying this equation to points 1 to 8 successively and using expressions (m) for the 
differences, we obtain the following system of linear equations: 

qb 2 
— lOikfi +4M 2 - - — 
9 

4Afi - 10A/ 2 + M z + 4M 4 - - — 

06 2 
M 2 - 103/, + 4Af s = - 2- 

9 



(P) 



4M 2 - lOM, + M t + 4M 7 = - — 

4M 3 + M< - 10M 6 + M 6 + 4Af 8 - - 2*! 

9 

3/ 8 - 10A/ 6 + 4A/ 7 = - 2^ 

9 

4Af 4 + 4M 6 - 10M, -}- Af 8 ^ 

9 

82lf s + 2M, - 10M 8 - - — 

9 

The solution of this system is 

ab 2 ah* 

Mi = 0.29942 — M 6 = 0.66191 — 
9 9 

M 2 = 0.49854 — Jlf 6 = 0.39387 — 
9 9 

</6 2 o6 2 

M 3 - 0.41462— M 7 = 0.56920 — 
9 9 

M A = 0.59329 ^ Ms = 0.74337 — 
9 9 

The second of the equations (n) now becomes 

Mb 1 
4A„w + A vy w = - — (r) 

Taking into account the result (q) this gives a second group of equations: 

-10wi +4w s = -0.29942AT 

4u>i - 10^2 + w z + 4w 4 = -0.49854W 

w 2 - 10w 3 + 4«; 6 = -0.41462AT 

4u> 2 - 10w 4 +w t + 4u> 7 = -0.59329JNT . . 

4u>, +w t - 10w 6 +w t + 4u> 8 = -0.66191iV W 

u> s - 10te« + 4u> 7 - -0.39387iV 

4u>< + 4u> 6 - 10u> 7 + w s = -0.56920AT 

8w s + 2u> 7 - 10u> 8 = -0.74337i\T 



(?) 



SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 359 



in which 



N « 



81D 



This yields the deflections 

wi = 0.13176JV 
w 2 = 0.25455AT 
w z = 0.22U1N 
w 4 = 0.32469AT 

It should be noted that the integration of 
the differential equation of the bent plate 
by analytic methods would encounter con- 
siderable difficulties in this case. 

To calculate the moments at the middle 
point 8 of the plate we have to use expres- 
sions (101) and (102), in which the deriva- 
tives first must be replaced by the respective 
differences. Thus, making use 1 of expres- 
sions (m) and using the values (t) for the 
deflections, and also taking v = 0.2, we 
obtain 



ws = 0.38549W 
w 6 - 0.20293iV 
wi = 0.31249JV 
ws = 0.44523iV 



(M x )s 
(M y )s 
(M xy ) s = (1 



-c 



Wi — 2wj + Wj 
Ax 2 

— 2m> 8 + W-! 



+ v 



+ p 



Fig. 183 

M>7 — 2w 8 + Wi 



Ay 2 

Wi — 2u>g + w b 



y)D 



Ay 2 
w t — Ws — Ws + Wa 



Ax 2 
= 0.0108g& 2 



)- 



0590g6 2 
0401<?6 2 



4Ax Ay 
Mohr's circle (Fig. 183) now gives 2 the following principal moments at point 8: 



M x + My , 



-*»* min — 



M x + M t 
2 



^(^^y + mi, - o.m^ 



+ M 2 = 0.0352</fo 2 



(0 



< ; M mQX 

— — M y — -H. 


S 


M n 


■4 Mrpjn ► 

^ . . . ... . , - M 


O^ 


M +X y 


«* ....... . — JVlx"- 

Mnt 









The direction of stresses due to these moments with respect to the coordinate axes 
x and y, respectively, is given by 



1 2M XV 
a = - arctan — ttt 

2 M X - My 



= 24°25' 



From Fig. 182 we conclude that the stresses due to M ma x at the center are acting almost 
exactly in the direction of the short span of the plate. 

The plan of the plate in Fig. 182 was such that we could use a rectangular network 

1 See also the diagrams in Fig. 184 for the particular case Ax = Ay. 

2 Note the difference of notations in Figs. 183 and 22. The principal moments in 
Fig. 183 are denoted by Af max and M m \ n . Note also that if in both diagrams the point 
on the circle moves in the clockwise direction, the normal to corresponding section 
will move in the same direction. 



360 



THEORY OP PLATES AND SHELLS 



with constant subdivisions Ax and Ay. In a more general case a triangular network 1 
must be used for the analysis of a skew slab. 

The method of finite differences can also be applied to plates with edges built in or 
free and, finally, to plates with mixed boundary conditions. 2 Since in the general 

© — ® — ® 

_ ••• 

ax ax 



— @ — 



>2 

o • • • 




A x* 



— ® — ® — @ — 



a 3 _iii. 

dx 3 2X 3 



0- 



D — ® — © 



<D 



© — ® — Q 



(-1) (03 



axdy 4X 



@ ® (D 



® ® @ 



© — ® — CD 
a 3 



axay 2 zx 3 



Ax = Ay.= X 
x 



®— -@ 



Fig. 184 



aA ... 
ax zx 3 



© 



® @ (2) 



I) — 




AA = T 



case the value of M is not fixed on the boundary, and accordingly the use of M becomes 
less advantageous, the deflections w may be calculated directly by means of a sequence 

1 Extensive use of such networks is made by V. P. Jensen in Univ Illinois Bull. 332, 
1941, and the previous numerical example is taken therefrom. 

2 Many numerical examples of this kind may be found in the book by H. Marcus, 
"Die Theorie elastischer Gewebe," 2d ed., Berlin, 1932; see also N. J. Nielsen, 
"Bestemmelse af Spaendinger i Plader," Copenhagen, 1920. 



SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 361 

of difference equations equivalent to the differential equation AAw = q/D of the bent 
plate. For convenience the finite difference equivalent of the operator AA( • • • ) is 
represented in Fig. 184 together with the other useful operators. The diagram is based 
on the assumption Ax = Ay = X. Each number has to be multiplied by the symbol Wk 
denoting the deflection at the respective point k and the sum of such products then 
divided by an expression given in the caption. 

In order to formulate the boundary conditions for an edge with vanishing deflections 
let us establish the equation for an interior point 7, next to the edge (Fig. 185). 
Applying the operator AA( • • • ) we have 

[wi + Wb + w 9 + Wit + 2(w 2 + Wa + wio + wi 2 ) 

1 Q7 
- 8(w 3 + We + Ws + Wn) + 20w 7 ) — = — (u) 

in which ty 2 = 1^3 = w A = 0. Next we have to eliminate the deflection Wi at a fictive 
point {, obtained by continuation of the network beyond the boundary of the plate. 







LU 


/ 


A 










/ 


5 2 ( 


,6 ( 


)'° 


A 

1 
f 

f 




, 


1 


,3 ( 


f v 

J ( 


)" c 


\ 

1 

>' 3 i 


~*i 


) v 


J \ 


) \ 

4 ( 


,8 ( 


) 12 


1 \ 

1 
<< 

1 

f 






Vj 


) v 
( 


,9 




1 




<--x~> 


<--x-> 


<— X— > 


) 


^-X-> 





Fig. 185 

This is readily done by means of the relation wi = -w 7 when the plate is simply sup- 
ported at point 3 and by means of wi = w 7 when the plate is built in. Thus, there 
remain only the deflections of the interior points in Eq. (u) and the total number of 
such unknown deflections will not exceed the number of the equations of the type (u) 
at our disposal. 

In the case of a free edge the number of such difference equations will be increased 
by the number of such points 2, 3, 4, . . . on the boundary at which the deflections 
do not vanish. The respective operators A Aw now must be extended over the exterior 
point at the distance X and also 2X from the free edge. Corresponding to each pair of 
such unknown deflections w , w i} there will be two boundary conditions 



d 2 w d 2 w 

— ~ + v — 7 = 
dx 2 dy 2 



~dx* dx dy 2 



= 



expressed by means of the differences and written for point 3, opposite to both exterior 
points and 1. Hence the total number of equations will still be the same as the 
number of unknown deflections. 

When the values of M in the interior of the plate are no longer independent of the 



362 THEORY OF PLATES AND SHELLS 

deflections w, the difference equations for the deflections become more involved than 
was the case in the two previous examples. In solving such equations the method of 
relaxation can sometimes be used to great advantage. 1 

84. Experimental Methods. For irregularly shaped plates or plates with irregularly 
varying thickness or weakened by many holes, experimental methods of investigation 
become more efficient than purely analytical methods. Conventional devices, such 
as electrical strain gauges and extensometers of all kinds, can be used for determination 
of strain in a bent plate. 2 The following brief review is restricted to methods which are 
appropriate to special conditions connected with the bending of thin elastic plates. 

Use of Photoelasticity. z This method, usually applied to problems of plane stress, 
must be necessarily altered if employed in the case of bending of plates. In fact, the 
normal stresses in a thin bent plate are equal in magnitude but opposite in sign for two 
fibers symmetrical with respect to the middle plane of the plate. Accordingly, the 
optical effect produced in the zone of tension on a beam of polarized light passing 
through the plate is nullified by an opposite effect due to the zone of compression. 

The influence of the second zone can be eliminated by cementing together two 
identical plates of photoelastic material with a reflecting foil of metal between them. 
The inner surface of one or both plates may also be silvered to the same end. 4 Calcula- 
tions show that the optical effect of such a sandwich plate of a thickness h is about the 
same as the effect of a single plate of the thickness h/2 if this latter plate is submitted 
to a plane stress equal to the extreme fiber stress of the bent plate. 

Another alternative 5 for making a bent plate photoelastically effective is to cement 
together two plates, both of photoelastic material, but having different elastic proper- 
ties. The law of distribution of the flexural stress is no longer linear in such a plate. 
Hence, being bent, it yields an optical effect on a beam of polarized light. 

According to a third method, sheets of photoelastic material are bonded on a reflec- 
tive surface of a plate of any elastic material and any dimensions. 6 The behavior of 
such sheets in a beam of polarized light yields all data regarding the strain in the 
extreme fibers of the tested plate. The method allows us to investigate the strain in a 

1 For this method, due to R. V. Southwell, see S. Timoshenko and J. N. Goodier, 
"Theory of Elasticity/' 2d ed., p. 468, New York, 1951. See also F. S. Shaw, "An 
Introduction to Relaxation Methods," Dover Publications, New York, 1953, where 
further bibliography is given. Another method of successive approximation in using 
the finite differences equation was developed by H. Liebman, Die angenaherte 
Ermittlung harmonischer Funktionen und konformer Abbildungen, Silzber. 
Munchen. Akad., p. 385, 1918. The convergency of this method was discussed by 
F. Wolf, Z. angew. Math. Mech., vol. 6, p. 118, 1926, and by R. Courant, Z. angew. 
Math. Mech., vol. 6, p. 322, 1926. For an improved method see also R. Zurmuhl, Z. 
angew. Math. Mech., vol. 37, p. 1, 1957. 

2 An electromechanical method in measuring curvatures of a bent slab was used by 
W. Andra, F. Leonhardt, and R. Krieger, Bauingenieur, vol. 33, p. 407, 1958. 

3 See for instance Timoshenko and Goodier, op. tit., p. 131. 

4 See J. N. Goodier and G. H. Lee, J. Appl. Mechanics, vol. 8, p. A-27, 1941, and 
M. Dantu, Ann. ponts et chaussies, p. 281, 1952. 

5 See H. Favre, Schweiz. Bauztg., 1950. For application of the method to a canti- 
lever plate of variable thickness see H. Schwieger and G. Haberland, Z. angew. Math. 
Mech., vol. 36, p. 287, 1956. 

6 This photostress method is due in principle to A. Mesnager (1930), but its practical 
application has been realized only recently; see, for example, F. Zandman and M. R. 
Wood, Prod. Eng., September, 1956. For application of the so-called freeze procedure 
to plates, see D. C. Drucker, J. Appl. Mechanics., vol. 9, p. A-161, 1942. 



SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 363 

slab which is part of an actual structure and subjected to the actual loading, rather 
than being restricted to a model of the slab. 

Use of Reflected Light. 1 The effect of a reflective surface of a strained plate on the 
direction of two adjacent light beams can be used to calculate the surface curvatures 
d 2 w/0x 2 } d 2 w/dy 2 , and d 2 w/dx dy, and, consequently, also the values of the flexural and 
torsional moments of the plate. For the same purpose the distortion of a luminous 
rectangular mesh projected on the initially plane surface of the plate may be used. 
Especially valuable are results obtained in this way for plates on elastic foundation, 
whose mechanical properties never can be expressed in a perfect manner analytically. 

The Interference Method. Similar to the classic method used for determination of 
Poisson's ratio on beams, the interference method has also been applied to measure 
the deflections of a bent plate. 2 

Analogy between Plane Stress and Plate Bending. 2 There is an analogy between the 
plate deflection, governed by the differential equation AAw = on the particular case 
of edge forces acting alone, and Airy 's stress function <p satisfying the equation AA*> = 0. 
Whereas the function w yields the curvatures of the deformed plate, Airy's function 
yields the components a x = d 2 <p/dy 2 ,a y = d 2 <p/dx 2 , and r xy = -d 2 <p/dx dy of the plane 
stress in an elastic solid. Provided the contour, say f(x,y) = 0, is the same in both 
cases, we can put 

d 2 w „ d 2 w Tjr d 2 w 

= K<j y = Kar x - — — = — &T xy 

dx 2 dy 2 dxdy 

where K is an arbitrary constant, such that the curvatures remain small. 

Measured deflections w can be used for computation of the components of the plane 
stress and vice versa if certain conditions of analogy are satisfied both on the boundary 
of the plate and on that of the elastic solid. 4 

1 For theory of the method and its application to various problems of bending of 
plates see M. Dantu, Ann. ponts et chausstes, 1940 and 1952. See also G. Bowen, 
Eng. News-Record, vol. 143, p. 70. 1949. 

2 See R. Landwehr and G. Grabert, Ingr.-Arch., vol. 18, p. 1, 1950. 

3 Established by K. Wieghardt, Mitt. Forschungsarb. Ingenieurwesens, vol. 49, 1908. 
For a further extension of the analogy see H. Schaefer, Abhandl. Braunschweig, wiss. 
Ges., vol. 8, p. 142, 1956. 

4 A simple formulation of those conditions is due to M. Dantu, Ann. ponts et 
chaussies, p. 386, 1952. For experimental methods based on analogy with electrical 
phenomena see R. H. MacNeal, J. Appl. Mechanics, vol. 18, p. 59, 1951, and K. 
Wotruba, Czechoslov. J. Phys., vol. 2, p. 56, 1953. Further information on various 
experimental methods may be found in L. Foppl and E. Monch, "Praktische Span- 
nungsoptik," 2d ed., Berlin, 1959. 



CHAPTER 11 

BENDING OF ANISOTROPIC PLATES 



85. Differential Equation of the Bent Plate. In our previous discus- 
sions we have assumed that the elastic properties of the material of the 
plate are the same in all directions. There are, however, cases in which 
an anisotropic material must be assumed if we wish to bring the theory 
of plates into agreement with experiments. 1 Let us assume that the 
material of the plate has three planes of symmetry with respect to its 
elastic properties. 2 Taking these planes as the coordinate planes, the 
relations between the stress and strain components for the case of plane 
stress in the xy plane can be represented by the following equations: 

<r x = E' x e x + E"e y 

*v = K*v + E " e * ( a ) 

Txy = Gjxy 

It is seen that in the case of plane stress, four constants, E' x , E' y} E h ', and 
G, are needed to characterize the elastic properties of a material. 

Considering the bending of a plate made of such a material, we assume, 
as before, that linear elements perpendicular to the middle plane {xy 
plane) of the plate before bending remain straight and normal to the 
deflection surface of the plate after bending. 3 Hence we can use our 
previous expressions for the components of strain: 

d 2 w d 2 w d 2 w /JA 

*--'5F *»=- z W 7 ** = ~ 2 **^ (6) 

1 The case of a plate of anisotropic material was discussed by J. Boussinesq, J. math., 
ser. 3, vol. 5, 1879. See also Saint Venant's translation of "Th^orie de r&asticit£ des 
corps solides," by A. Clebsch, note 73, p. 693. 

* Such plates sometimes are called "orthotropic." The bending of plates with more 
general elastic properties has been considered by S. G. Lechnitzky in his book "Aniso- 
tropic Plates/' 2d ed., Moscow, 1957. 

3 The effect of transverse shear in the case of anisotropy has been considered by 
K. Girkmann and R. Beer, Osterr. Ingr.-Arch., vol. 12, p. 101, 1958. 

364 



BENDING OF ANISOTROPIC PLATES 365 

^he corresponding stress components, from Eqs. (a), are 

»---(*£ E +'*S?) 



Tij, = — 26r2 



3x 3t/ 
Vith these expressions for stress components the bending and twisting 



aoments are 



fh/2 / d 2 w . _. a 2 uA 

^-/>*--W + ^Sf) (212) 



-A/2 



M Iy = - / t xj/ 2 dz = 2D iy ^-^ 



n which 



n E#» n E'Jk* n _E"K> _Gh* 

D* = ~J2 ^--fr Dl ""l2" "-""12 (d) 

Substituting expressions (212) in the differential equation of equilibrium 
100), we obtain the following equation for anisotropic plates: 

D ,0 + 2( Z> 1 + 2D „)^ + ^ = , 

introducing the notation 

H = Di + 22)^ (e) 

,ve obtain 

-rv d A w , ^ TT d 4 w , t^ d A w /010 v 

i> -3^ + 2H teW + Z> -V =? (213) 

The corresponding expressions for the shearing forces are readily obtained 
'rom the conditions of equilibrium of an element of the plate (Fig. 48) 
md the previous expressions for the moments. Thus, we have 

n d (n d * w 4- H d * w \ 

[n the particular case of isotropy we have 

K-E,- j-— - 2 & - x _ y2 (x 2(1 + y) 



(/) 



366 THEORY OF PLATES AND SHELLS 

Hence 

Eh' 
D x - D y - 12(1 _ ^ 

on W ( vE , E \ __ Eh' 

H = £>i + 2L>,„ - ^2 \Y^r? + i + v ) - 12(1 - v 2 ) 

and Eq. (213) reduces to our previous Eq. (103). 

Equation (213) can be used in the investigation of the bending of plates 
of nonisotropic and even nonhomogeneous material, such as reinforced 
concrete slabs, 1 which has different fiexural rigidities in two mutually 
perpendicular directions. 

86. Determination of Rigidities in Various Specific Cases. The expressions (d) 
given for the rigidities in the preceding article are subject to slight modifications accord- 
ing to the nature of the material employed. In particular, all values of torsional 
rigidity D xy based on purely theoretical considerations should be regarded as a first 
approximation, and a direct test as shown in Fig. 25c must be recommended in order 
to obtain more reliable values of the modulus G. Usual values of the rigidities in some 
cases of practical interest are given below. 

Reinforced Concrete Slabs. Let E a be Young's modulus of steel, E c that of the con- 
crete, v c Poisson's ratio for concrete, and n = EJE C . In terms of th e elastic con- 
stants introduced in Art. 85 we have approximately v c = E"/y/Ef x E> v . For a slab 
with two-way reinforcement in the directions x and y we can assume 

D x - — ^- 2 [7„ + (n - 1)/.,] 

D y - — ^ [/,„ + (n - 1)/,,] (n . 

1 — v" W 



2>i = v c \/D x D y 

1 — Vc / 

D X y = " VD X Dy 

In these equations, J c * is the moment of inertia of the slab material, I 9X that of the 
reinforcement taken about the neutral axis in the section x = constant, and I ey and 
I av are the respective values for the section y = constant. 

With the expression given for D xy (also recommended by Huber) we obtain 



H - VD X D U (b) 

and the differential equation 

1 The application of the theory of anisotropic plates to reinforced concrete slabs is 
due to M. T. Huber, who published a series of papers on this subject; see Z. Osterr. 
Ing. u. Architektur Ver., 1914, p. 557. The principal results are collected in his books: 
"Teorya Ply t," Lvov, 1922, and "Probleme der Statik technisch wichtiger orthotroper 
Platten," Warsaw, 1929. Abstracts of his papers are given in Compt. rend., vol. 170, 
pp. 511 and 1305, 1920; and vol. 180, p. 1243, 1925. 



BENDING OF ANISOTROPIC PLATES 



367 



hich can readily be reduced to the form (103) by introducing 2/1 = y -\/D x /Dy as a 
ew variable. 

It is obvious that the values (a) are not independent of the state of the concrete, 
or instance, any difference of the reinforcement in the directions x and y will affect 
le ratio D x /D y much more after cracking of the concrete than before. 

Plywood. For a plate glued together of three or five plies, the x axis supposed to be 
arallel to the face grain, we may use the constants given in Table 79. 

Table 79. Elastic Constants for Plywood 
Unit = 10 6 psi 



Material 



laple,* 5-ply 

fara,* 3-ply 

iaboon* (OkoumS), 3-ply 

;irch,f 3- and 5-ply 

iircht with bakelite membranes. 



e: 



1.87 
1.96 
1.28 
2.00 
1.70 



EL 



0.60 

0.165 

0.11 

0.167 

0.85 



E" 



0.073 
0.043 
0.014 
0.077 
0.061 



G 



0.159 

0.110 

0.085 

0.17 

0.10 



* By R. F. S. Hearmon and E. H. Adams, Brit. J. Appl. Phys., vol. 3, p. 155, 1952- 
t By S. G. Lechnitzky, "Anisotropic Plates," p. 40, Moscow, 1947. 




1 

I 1 i 
i i 1 ! 

11' 


v 


1.-1-J 



Fig. 186 

Corrugated Sheet. Let E and v be the elastic constants of the material of the sheet, 
its thickness, 



z = / sin 



I 



he form of the corrugation, and s the length of the arc of one-half a wave (Fig. 186). 
?hen we have 1 

I Eh' 



B x = 



s 12(1 - v 2 ) 



Dy = EI 

Di ~0 
H = 2D xy = 



Eh* 



I 12(1 + v) 
1 See E. Seydel, Ber. deut. Versuchsanstalt Luftfahrt, 1931, 



368 



THEORY OF PLATES AND SHELLS 



in which, approximately, 



-(-3) 



'-? 



1 - 



0.81 



1 +2 



4)'J 



Piate Reinforced by Equidistant Stiffeners in One Direction. For a plate reinforced 
symmetrically with respect to its middle plane, as shown in Fig. 187, we may take 1 



D x - H = 



£A» 



12(1 
Eh* 



+ 



£7 



12(1 - **) ax 

in which E and * are the elastic constants of the material of the plating, E' the Young 
modulus, and / the moment of inertia of a stiffener, taken with respect to the middle 
axis of the cross section of the plate. 



I 1 



Jb. 



IT 



CT 



! * i 

|<- Qi — *| ->|t 



f± 



Fig. 187 

Plate Cross-stiffened by Two Sets of Equidistant Stiffeners. 
ment is still symmetrical about the plating we have 



Fig. 188 
Provided the reinforce- 



D x = 

H - 



Eh* 



12(1 - „*) 



+ 



Eh* 



+ ■ 



E'l, 



12(1 - v 8 ) a, 

Jgft* 
12(1 - K») 



Ji being the moment of inertia of one stiffener and 6i the spacing of the stiffeners in 
direction x, and I2 and a t being the respective values for the stiffening in direction y. 

Slab Reinforced by a Set of Equidistant Ribs. In the case shown in Fig. 188 the 
theory established in Art. 85 can give only a rough idea of the actual state of stress and 

1 Recommended by Lechnitzky, op. cit. For more exact values see N. J. Huffington, 
J. Appl. Mechanics, vol. 23, p. 15, 1956. An experimental determination of the 
rigidities of stiffened and grooved plates was carried out by W. H. Hoppmann, N. J. 
Huffington, and L. S. Magness, J. Appl. Mechanics, vol. 23, p. 343, 1956. 



BENDING OF ANISOTROPIC PLATES 



369 



;rain of the slab. Let E be the modulus of the material (for instance, concrete), 
the moment of inertia of a T section of width a u and a = h/H. Then we may 

ssume 

Eaji 3 

Dx = 12(oi - t + aH) 

EJ_ 

ai 
D x = 

Tie effect of the transverse contraction is neglected in the foregoing formulas. The 
Drsional rigidity, finally, may be calculated by means of the expression 



Dy = — 



D x 



D' xy + 



2a, 



i which D' is the torsional rigidity of the slab without the ribs and C the torsional 

xy 

igidity of one rib. 1 

87. Application of the Theory to the Calculation of Gridworks. Equa- 
ion (213) can also be applied to the gridwork system shown in Fig. 189. 




Fig. 189 
This consists of two systems of parallel beams spaced equal distances 
apart in the x and y directions and rigidly connected at their points of 
intersection. The beams are supported at the ends, and the load is 
applied normal to the xy plane. If the distances a x and bx between the 
beams are small in comparison with the dimensions a and b of the grid, 
and if the flexural rigidity of each of the beams parallel to the x axis is 
equal to Bi and that of each of the beams parallel to y axis is equal to B 2 , 
we can substitute in Eq. (213) 



D =^ 



ai 



(a) 



i For a more exact theory concerning slabs with ribs in one or two directions and 
leading to a differential equation of the eighth order for the deflection see K. Trenks, 
Bauingenieur, vol. 29, p. 372, 1954; see also A. Pfliiger, Ingr.-Arch., vol. 16, p. Ill, 1947. 



370 



THEORY OF PLATES AND SHELLS 



The quantity D\ in this case is zero, and the quantity D xy can be expressed 
in terms of the torsional rigidities Ci and C 2 of the beams parallel to the 
x and y axes, respectively. For this purpose we consider the twist of an 
element as shown in Fig. 1896 and obtain the following relations between 
the twisting moments and the twist d 2 w/dx dy: 



M xv = P 



Ci d*w 



xy 



bi dx dy 



M yx = 



Co d 2 w 



a\ dx dy 



GO 



Substituting these expressions in the equation of equilibrium (e) on page 
81, we find that in the case of the system represented in Fig. 189a the 
differential equation of the deflection surface is 






+ 



/Ci CA d*w , Bj 

\bi + at) dx 2 dy* "•" a x 



d 4 w 



(215) 



which is of the same form as Eq. (213). 



In order to obtain the final expressions for the flexural and torsional moments of a 
rib we still have to multiply the moments, such as given by Eqs. (212) and valid for 
the unit width of the grid, by the spacing of the ribs. The variation of the moments, 
say M x and M xy , may be assumed parabolic between the points (m — 1) and 
(m + 1) and the shaded area of the diagram (Fig. 190) may be assigned to the rib 





< fy > 




M m -i 


Mm 


M m +i 


m-1 
<_ t 


m 
>|< b 1 -> 


m+1 



Fig. 190 



(m) running in the direction x. Then, observing the expressions (212), we obtain the 
following approximate formulas for both moments of the rib (m): 



M 3 



M x 



24 L\ax s /„,-, \da. 2 / m \dx*J m+ i] 

24 |_ \dx dy/ n -i \dx dy/ m \dx dy/ m+ i J 



(c) 



For ribs of the direction y we have to interchange x and y in the foregoing expressions 
and replace Bi by B 2 and C, by d; (m - 1), (m), and (m + 1) then denote three 
successive joints on a rib having the direction x. 



BEXDTNG OF ANISOTROPIC PLATES 371 

Two parameters largely defining the elastic properties of a grid and often used in 
calculation are 

C\ C2 

B 2 bi _ 1 bi a\ 

B\0,i 2 ~ ~ 



-i 



[BJh 
\ai6, 



The parameter X multiplied by the side ratio a/b (Fig. 189) yields the relative carrying 
capacity of a rectangular plate in the directions y and x, whereas the parameter p 
characterizes the torsional rigidity of a grid as compared with its flexural rigidity. 

Equation (215) has been extensively used in investigating the distribution of an 
arbitrarily located single load between the main girders of a bridge stiffened in the 
transverse direction by continuous floor beams. 1 

88. Bending of Rectangular Plates. When the plate is simply sup- 
ported on all sides Eq. (213) can be solved by the methods used in the 
case of an isotropic plate. Let us apply the Navier method (see Art. 28) 
and assume that the plate is uniformly loaded. Taking the coordinate 
axes as shown in Fig. 59 and representing the load in the form of a double 
trigonometric series, the differential equation (213) becomes 



D 



d 4 w 
dx* 


+ 2H 


d 4 w 
dx 2 dy z 


+ D d * W 
+ v dy< 

16g 

7T 2 



00 00 

2 I 

m =1,3,5,... n= 1,3,5, 



1 . rrnrx . 1VKV , N 

— sin sin ——- (a) 

mn a b 



A solution of this equation that satisfies the boundary conditions can be 
taken in the form of the double trigonometrical series 

00 00 

XV* . rrnrx . rnry /1A 

} a mn sin sin -j— (b) 

m= 1,3,5, ... n = 1,3,5, ... 

Substituting this series in Eq. (a), we find the following expression for the 
coefficients a mn : 

_ lQqo 1 

dmn 



7T 6 






mn 



1 Factors giving the distribution of a single load have been calculated for /* = by 
Y. Guyon, Ann. ponts et chaussbes, vol. 116, p. 553, 1946, and for p j± by C. Mas- 
sonnet, Pubis. Intern. Assoc. Bridge and Structural Engrs., vol. 10, p. 147, 1950. For 
verification of calculated results by test see K. Sattler, Bauingenieur, vol. 30, p. 77, 
1955, and also M. Naruoka and H. Yonezawa, Pubis. Intern. Assoc. Bridge and 
Structural Engrs., vol. 16, 1956. For skewed grids see S. Woinowsky-Krieger, Ingr.- 
Arch., vol. 25, p. 350, 1957. 



372 THEORY OF PLATES AND SHELLS 

Hence the solution of Eq. (a) is 



w = 



. rrnrx . mry 

^ ^ ^ ^ sin sin -r- 

16g \ A \ A ^_ 



^-^ ^"^ ran 

m = 1,3,5, ... n = 1,3,5, ... 



/V 2m«n» n* \ 



(c) 



In the case of an isotropic material D x = D v = H = D, and this solution 

coincides with that given on page 110. 

Furthermore, let us consider the particular case of H = -\/D x D u 
already mentioned on page 366. Comparing expression (c) with the 
corresponding expression (131) for the isotropic plate, we conclude that 
the deflection at the center of such an orthotropic plate with rigidities 
D x , D y , and the sides a, b is the same as th at of an isotro pic pla te having 
a rigidity D and the sides a = a \/D/D x and b = b -\/D/D y . In like 
manner the curvatures of the orthotropic plate may be expressed by 
those of a certain isotropic plate. The deflection and the bending 
moments at the center of the orthotropic plate obtained in this way 
can be expressed by the formulas 



gob 4 



/ v n [7T\ 



«*' (<J) 



where a, 0i, and j3 2 are numerical coefficients 1 given in Table 80 and 

As a second example let us consider an infinitely long plate (Fig. 74) 
and assume that the load is distributed along the x axis following the 
sinusoidal relation 

q = go sin \J) 

a 

In this case Eq. (213) for the unloaded portions of the plate becomes 

1 Calculated by M. T. Huber, "Probleme der Statik technisch wichtiger orthotroper 
Platten," p. 74, Warsaw, 1929. For numerical data regarding uniformly loaded 
rectangular plates with various edge conditions and various torsion coefficients, see 
H. A. Schade, Trans. Soc. Naval Architects Marine Engrs., vol. 49, pp. 154, 180, 1941. 



BENDING OF ANISOTROPIC PLATES 



373 



Table 80. Constants a, fi h and /3 2 for a Simply Supported Rectangular 
Orthotropic Plate with H = \/D x Dy, Eqs. (d), (e) (Fig. 59) 



e 


a 


0i 


02 


t 


a 


0i 


02 


1 


0.00407 


0.0368 


0.0368 


1.8 


0.00932 


0.0214 


0.0884 


1.1 


0.00488 


0.0359 


0.0447 


1.9 


0.00974 


0.0191 


0.0929 


1.2 


0.00565 


0.0344 


0.0524 


2.0 


0.01013 


0.0174 


0.0964 


1.3 


0.00639 


0.0324 


0.0597 


2.5 


0.01150 


0.0099 


0.1100 


1.4 


0.00709 


0.0303 


0.0665 


3 


0.01223 


0.0055 


0.1172 


1.5 


0.00772 


0.0280 


0.0728 


4 


0.01282 


0.0015 


0.1230 


1.6 


0.00831 


0.0257 


0.0785 


5 


0.01297 


0.0004 


0.1245 


1.7 


0.00884 


0.0235 


0.0837 


00 


0.01302 





0.1250 



A solution of this equation, satisfying the boundary conditions at the 
sides parallel to the y axis, can be taken in the following form: 



w = Ym sin 



rrnrx 
a 



(h) 



where Y m is a function of y only. Substituting this in Eq. (g), we obtain 
the following equation for determining the function Y m : 



D Y iy 



2H 



ra 2 7r 2 



Yl l + D, 



Y. 







a z a" 

The roots of the corresponding characteristic equation are 



_ mir JH IIP D: 

r llM> 4- ±-^yiTjr v ±^ D i Dy 

Using, in accordance with Eq. (d), Art. 87, the notation 

* - 4 /5 _ H 



(0 



u) 



w 



'x-LSy 



we have to consider the following three cases: 
Case 1, m > 1- 



Case 2, ju = 1 - 
Case 3, /x < 1 : 



H 2 > D x D y 
H 2 = D.D^ 
i/ 2 < D X Z>, 



(0 



In the first case all the roots of Eq. (j) are real. Considering the part 
of the plate with positive y and observing that the deflection w and its 
derivatives must vanish at large distances from the load, we can retain 



374 THEORY OF PLATES AND SHELLS 

only the negative roots. Using the notation 



7T 



= ^ V M - V7^~i 



(w) 



the integral of Eq. (i) becomes 

Y m = A m e- m vi° + B m er«*i* 
and expression (h) can be represented in the form 

w = (A m e- mvla + B m e- mvlfi ) sin — — 

(JL 

From symmetry Ave conclude that along the x axis 



c 



and we find 

B ~ -^A 

a 

and w = A m ( e~ myla — - er mvl * J sin ^^ (n) 

\ a J a 

The coefficient 4 m is obtained from the condition relating to the shearing 
force Q y along the x axis, which gives 

d ( ~ d 2 w . TT d 2 w\ q . m7r.r 

Substituting for w its expression (n), we obtain 

a _ <7o<2 3 /3 2 __ ag a 4 



2m 3 D y (a 2 - 2 ) 2ir i m 3 D x (a i - /3 2 ) 
and the final expression (w) for the deflection becomes 

• " 2*.W - W (aC -""" " ^' 8) Si " T ( °> 

In the second of the three cases (I) the characteristic equation has two 
double roots, and the function Y m has the same form as in the case of an 
isotropic plate (Art. 36). In the third of the cases (l) we use the notation 



, a\ I 2 

a = vyr^ 

P TT \l-f M 



<P) 



BENDING OF ANISOTROPIC PLATES 375 

and thus obtain the solution 

«2 



qoa' 

w - 



/ , . my , n , my\ ,„, . rmrx 
, =: ( a' sin -7 + j8' cos — f ) e~ mxie sin ( Q ) 

We can also shift from case 1 to case 2 by using the complex relations 

1 = 1 _ 1 

« f V (r) 

1 1 .J_ ^ } 

Having the deflection surface for the sinusoidal load (/), the deflection 
for any other kind of load along the x axis can be obtained by expanding 
the load in the series 



00 

X. mir: 
a m sin 
a 



mirx 



and using the solution obtained for the load (/) for each term of this 
series. The following expressions hold when, for instance, a load P is 
concentrated at a point x = £, y = of the infinite strip (Fig. 72) : 

Case 1, m > 1- 

00 

Pa* 1 V 1 / / ^ ian • m nZ • ™>Trx , N 
w = -T^~ — ^ > -r, (ae-" 1 *'" - /Se"™^) sin sin (s) 

m = l 

Case 2, ju = 1 : 



= 7 ' a °~ V JL ^i i m7r ^ -»^ ~« m7r * ~- ^ 
27r 3 i) x X Li m 3 \ 



^ = ^-r^ > ^ ( 1 + ^fj <r-"" sin — sin — (0 

m= 1 



Case 3, \i < 1 : 

oo 

y°a V^ i / . m/ y 77?, A ,„, . rrnr£ . m7r.r 

™ = o"T'"~7TrTr / — ^ ( a sin ~t + i 8 cos — r ) e ~ mW ^ sin — i sin 

2tt 2 VD x D u Z^m 3 \ a' a' / a a 

ra= 1 

(U) 

Expressions in closed form 1 can be obtained for bending moments due to 
a single load in a manner similar to that used for the isotropic plate in 
Art. 35. 

Having this solution, the deflection of the plate by a load distributed 

1 See W. Nowacki, Acta Tech. Acad. Sci. Hung., vol. 8, p. 109, 1954; S. Woinowsky- 
Krieger, Ingr.-Arch., vol. 25, p. 90, 1957. Numerical results regarding influence 
surfaces of orthotropic rectangular plates may be found in H. Olsen and F. Reinitz- 
huber, "Die zweiseitig gelagerte Platte," Berlin, 1950, and in H. Homberg and 
J. Wcinmeistcr, "Einflussflachen fur Kreuzwerke," 2d cd., Berlin, 1956. 



376 THEORY OF PLATES AND SHELLS 

over a circular area can be obtained by integration, as was shown in the 
case of an isotropic plate (see Art. 35). By applying the method of 
images the solutions obtained for an infinitely long plate can be used in 
the investigation of the bending of plates of finite dimensions. 1 

89. Bending of Circular and Elliptic Plates. A simple solution of Eq. (213) can be 
obtained in the case of an elliptic plate clamped 2 on the boundary and carrying a 
uniform load of intensity q. Provided the principal directions x and y of the ortho- 
tropic material are parallel to the principal axes of the ellipse (Fig. 157) the expression 



/ * 2 y 2 V 



W = Wo ( 1 - — - To I (<*) 

in which 

W ° = 24£>x 16# 24JP y ^ 

a* + a 2 6 2 + 6 4 

satisfies Eq. (213) and the required conditions on the boundary. The bending 
moments of the plate are readily obtained by means of expressions (212). In the 
particular case of a clamped circular plate (a = b) we have the following results: 

o(a 2 - r 2 ) 2 

w = 

642)' 

M x - ^ [(£>* + I>i)(a 2 - r 2 ) - 2(A c s 2 + D l2/ 2 )] 



in which 



M « - 7^ KJ>» + A) (a* - r») - 2(D v y* + D**)] 
loD 

M ' v = 4D' DxyXV 

Q ' = ~ ^ (3Dx + H) 

Qv= ~8D' (3Dv + H) 



= Vx 2 + y l and D' = %(3D X + 2H + 3Z>„) 



(c) 



Since the twist is zero along the edge, the reactions of the support are given by a linear 
combination of the boundary values of the shearing forces Q x and Q v (see page 87). 
A straightforward solution can also be obtained in the case of pure bending or pure 
twist of an orthotropic plate. Let such a plate be subjected to uniform couples 
M x = Mi, M v = M 2 , and M xy = M 3 . By taking the deflection in the form 

w - Ax 2 + Bxy + Cy 2 (d) 

1 Several examples of this kind are worked out in the books by M. T. Huber: 
"Teorya Plyt," Lvov, 1922, and "Probleme der Statik technisch wichtiger orthotroper 
Platten," Warsaw, 1929. 

2 For bending of a simply supported elliptical plate, see Y. Ohasi, Z. angew. Math. u. 
Phys., vol. 3, p. 212, 1952. 



RENDING OF ANISOTROPIC PLATES 377 

we obviously satisfy the differential equation (213). The constants A, JS, and Cthen 
are given by the linear equations 

D X A + DiC = -£Mi 

D X A + D y C = ~iM 2 (e) 

D xy B = \M 3 

which ensue from the expressions (212). 

The bending of a circular plate with cylindrical aeolotropy has been discussed too. 1 
If, in addition to the elastic symmetry, the given load distribution is also symmetrical 
about the center of the plate, then the ordinary differential equation of the bent plate 
contains only two flexural rigidities, the radial and the tangential. Formal solutions 
of this equation for any boundary conditions are simple to obtain; the choice of the 
elastic constants of the material, however, requires special consideration since certain 
assumptions regarding these constants lead to infinite bending moments at the center 
of the plate even in the case of a continuously distributed loading. 

Most of the special methods used in solving the problems of bending of an isotropic 
plate (Chap. 10) can be applied with some modifications to the case of an anisotropic 
plate as well. 

If we take the complex variable method, 2 for example, the form of the solution 
proves to be different from that considered in Art. 79. As can be shown, it depends 
upon the roots pi, p 2 , — pi, and — p 2 of the characteristic equation 

D yP A + 2H P 2 + D x = 

which are either imaginary or complex. These roots being determined, the solution 
of the homogeneous equation D x d A wJdx A + 2H d 4 wi/dx 2 dy 2 + D v d 4 Wi/dy 4 = can 
be represented either in the form 

wi = (R[<pi(zi) + <p 2 (z?)} 

if Pi 5* P2, or else in the form 

wi = (R[<pi(zi) + zi<p 2 (zi)] 

if pi = p 2 . In these expressions <p\ and <p 2 are arbitrary analytic functions of the com- 
plex variables Z\ = x + p\V and z 2 = x + p*y. 

In using the Ritz method, expression (6) of Art. 80 for the strain energy has to be 
replaced by the expression 

T r 1 f a f b T ^ /d 2 w\ 2 ^ d 2 w d 2 w ^ [d 2 w\ ^ ( d*w VI ,, 7 

while the rest of the procedure remains the same as in the case of the isotropic plate. 

1 G. F. Carrier, /. Appl. Mechanics, vol. 11, p. A-129, 1944, and Lechnitzky, op. cit. 

2 See S. G. Lechnitzky, Priklad. Mat. Mekhan., vol. 2, p. 181, 1938, and V. Morcovin, 
Quart. Appl. Math., vol. 1, p. 1 16, 1943. For application of the method to the problem 
of stress concentration, see also G. N. Savin, "Stress Concentration around Holes/' 
Moscow, 1951, and S. G. Lechnitzky, Inzhenernyi Sbornik, vol. 17, p. 3, 1953. Stress 
concentration in isotropic and anisotropic plates was also discussed by S. Holgate, 
Proc. Roy. Soc. London, vol. 185A, pp. 35, 50, 1946. 



CHAPTER 12 

BENDING OF PLATES UNDER THE COMBINED ACTION 

OF LATERAL LOADS AND FORCES IN THE MIDDLE 

PLANE OF THE PLATE 



90. Differential Equation of the Deflection Surface. In our previous 
discussion it has always been assumed that the plate is bent by lateral 
loads only. If in addition to lateral loads there are forces acting in the 
middle plane of the plate, these latter forces may have a considerable 
effect on the bending of the plate and must be considered in deriving the 
corresponding differential equation of the deflection surface. Proceed- 
ing as in the case of lateral loading (see Art. 21, page 79), we consider 
the equilibrium of a small element cut from the plate by two pairs of 
planes parallel to the xz and yz coordinate planes (Fig. 191). In addi- 
tion to the forces discussed in Art. 21 we now have forces acting in the 

middle plane of the plate. We denote the 
magnitude of these forces per unit length by 
N x , N y , and N xy = N yx , as shown in the figure. 
Projecting these forces on the x and y axes 
and assuming that there are no body forces or 
tangential forces acting in those directions at 
the faces of the plate, we obtain the following 
equations of equilibrium: 




ax 

dNx 






dN x 

dx 

dN 



+ 



dN 



XII 



xu 



dx 



+ 



By 

BNy 

By 



= 



= 



(216) 



X Ny + 



aN v 

dy 



dy 



(b) 



Fig. 191 



These equations are entirely independent of 

the three equations of equilibrium considered 

in Art. 21 and can be treated separately, as 

will be shown in Art. 92. 

In considering the projection of the forces shown in Fig. 191 on the 

z axis, we must take into account the bending of the plate and the 

resulting small angles between the forces N x and N v that act on the 



opposite sides of the element. 



As a result of this bending the projection 
378 



PLATES UNDER LATERAL LOADS AND FORCES IN THEIR PLANE 379 

of the normal forces N z on the z axis gives 

tit j dw , /*r . dN x , \ {dw , d 2 w , \ , 

- w - "» SF + (*■ + "ST *y (s? + a? ^'j * 

After simplification, if the small quantities of higher than the second 
order are neglected, this projection becomes 

d 2 w dN x dw 

N x — 2 dx dy + -^-^ dx dy (a) 

In the same way the projection of the normal forces N y on the z axis gives 



d 2 w 7 7 , dN y dw 
W- dxd ' J + ^lTy 



^2/ "FT? d - c rf # + ~^7" "^7 d;t " ^ '&) 



Regarding the projection of the shearing forces N xy on the 2 axis, we 
observe that the slope of the deflection surface in the y direction on the 
two opposite sides of the element is dw/dy and dw/dy + (d 2 w/dx dy) dx. 
Hence the projection of the shearing forces on the z axis is equal to 

at d2w j a 1 dN *y dw A J 
N xy dx dy + -5—^ — dx dy 

dx dy dx dy 

An analogous expression can be obtained for the projection of the shear- 
ing forces N yx = N XI/ on the z axis. The final expression for the projec- 
tion of all the shearing forces on the z axis then can be written as 

oat d2w a a 1 dN xy dw dNxydw, , f v 

2N** ^— ^- da; dy + -3-^ -5— dx dy + -^ — da; dy (c) 

dx dy dx dy dy dx 

Adding expressions (a), (6), and (c) to the load q dx dy acting on the ele- 
ment and using Eqs. (216), we obtain, instead of Eq. (100) (page 81), the 
following equation of equilibrium: 



1 dx dy + dy' \ q + " x dx* + "" dy* + * N " dx dy J 



aw, 
ax 2 

Substituting expressions (101) and (102) for M x , M y , and M xu , we obtain 



d*w d 4 w dha 

dx 4 + ax 2 ay 2 + al/ 1 



i / . A7 . a 2 w . A7 . a 2 w . OA , a 2 w \ , _. 
5 ( 9 + *' ^ + *" a^ + 2Nxy dx-Yy) (217) 

This equation should be used instead of Eq. (103) in determining the 
deflection of a plate if in addition to lateral loads there are forces in the 
middle plane of the plate. 



380 THEORY OF PLATES AND SHELLS 

If there are body forces 1 acting in the middle plane of the plate or tangential forces 
distributed over the surfaces of the plate, the differential equations of equilibrium 
of the element shown in Fig. 191 become 

5^+5^ + x-o 

dx dy (218) 

dx dy 

Here X and Y denote the two components of the body forces or of the tangential 
forces per unit area of the middle plane of the plate. 

Using Eqs. (218), instead of Eqs. (216), we obtain the following differential equa- 
tion 2 for the deflection surface: 



d 4 w „ d A w , d 

h 2 h - 

dx 4 dx 2 dy 2 dy 



4 W 1 / d 2 W d 2 W , d 2 W w dW „ dw\ 

y 4 Dy dx 2 v dy 2 v dxdy dx dy J 



(219) 



Equation (217) or Eq. (219) together with the conditions at the boundary (see Art. 22, 

page 83) defines the deflection of a plate loaded 
•X laterally and submitted to the action of forces in the 
middle plane of the plate. 





r*n 1 »i i 

«- -+ b 

— -* S 

U— ■ H — L 

* a > 



91. Rectangular Plate with Simply Supported 
Edges under the Combined Action of Uniform 
Lateral Load and Uniform Tension. Assume 
y that the plate is under uniform tension in the 

Fig. 192 x ( jj rect j on ^ as s hown in Fig. 192. The uniform 

lateral load q can be represented by the trigonometric series (see page 109). 

oo eo 

16a V V* 1 . rmrx . nwy , * 

q = — f- > / — sin sin -r 2 - (a) 

m = 1,3,5,... n = 1,3,5,... 

Equation (217) thus becomes 

d 4 w d 4 w d^w _ N x d 2 w 

dx 4 + dx 2 dy 2 + dy 4 D dx 2 



16? V V 

Dtt 2 Li L< 



rmrx . n7rt/ 



— sin sin — r^- (o) 

mn a o 

m= 1,3,5, ... n = 1,3,5, 

This equation and the boundary conditions at the simply supported edges 

1 An example of a body force acting in the middle plane of the plate is the gravity 
force in the case of a vertical position of a plate. 

2 This differential equation has been derived by Saint Venant (see final note 73) in 
his translation of Clebsch, "Th<§orie de l'elasticit6 des corps solides," p. 704, 1883. 



PLATES UNDER LATERAL LOADS AND FORCES IN THEIR PLANE 381 

will be satisfied if we take the deflection w in the form of the series 

V* \^ • ™>vx . rnry , N 

w = 2j/j amn Sin ~a~ Sm ~b~ 

Substituting this series in Eq. (6), we find the following values for the 
coefficients a mn : 

a = ^L— = (d) 



Dir*mn 



(WL 4- ZL*Y 4- iV ^ m2 

\a* + b*J t*Do* 



in which ra and n are odd numbers 1, 3, 5, ... , and a mn = if m or n 
or both are even numbers. Hence the deflection surface of the plate is 



00 00 

-a 2 X 

m= 1,3,5, . . . n = 1,3,5... 



|_\a 2 + 6 2 / + 7r 2 Z)a 2 J 



sin sm — r- \ e ) 

a b 



mn 



Comparing this result with solution (131) (page 110), we conclude from 
the presence of the term N x m 2 /ir 2 Da 2 in the brackets of the denominator 
that the deflection of the plate is somewhat diminished by the action of 
the tensile forces N z . This is as would be expected. 

By using M. Levy's method (see Art. 30) a solution in simple series 
may be obtained which is equivalent to expression (e) but more con- 
venient for numerical calculation. The maximum values of deflection 
and bending moments obtained in this way 1 for v = 0.3 can be represented 
in the form 



Wmax = OL 



qb* 



(M,) max = Pqb* (M y ) m „ = frqb* (/) 



The constants a, /3, and /3i depend upon the ratio a/b and a parameter 



y = 



4tt 2 JD 



and are plotted in Figs. 193, 194, and 195. 

If, instead of tension, we have compression, the force N z becomes 

1 H. D. Conway, /. Appl. Mechanics, vol. 16, p. 301, 1949, where graphs in the case 
of compression are also given; the case N x = N y has been discussed by R. F. Morse 
and H. D. Conway, J. Appl. Mechanics, vol. 18, p. 209, 1951, and the case of a plate 
clamped all around by C. C. Chang and H. D. Conway, J. Appl. Mechanics, vol. 19, 
p. 179, 1952. For combined bending and compression, see also J. Lockwood Taylor, 
The Shipbuilder and Marine Engine Builder, no. 494, p. 15, 1950. 



382 



THEORY OF PLATES AND SHELLS 



negative, and the deflections (e) become larger than those of the plate 
bent by lateral load only. It may be seen also in this case that at cer- 
tain values of the compressive force N x the denominator of one of the 
terms in series (e) may vanish. This indicates that at such values of N x 
the plate may buckle laterally without any lateral loading. 

92. Application of the Energy Method. The energy method, which 
was previously used in discussing bending of plates by lateral loading 
(see Art. 80, page 342), can be applied also to the cases in which the 



0.14 



0.12 



0.10 



—•"^ ^^*a 



t 



0.08 



0.06 



0.04 



0.02 



o 
b 

Fig. 193 

lateral load is combined with forces acting in the middle plane of the 
plate. To establish the expression for the strain energy corresponding 
to the latter forces let us assume that these forces are applied first to the 
unbent plate. In this way we obtain a two-dimensional problem which 
can be treated by the methods of the theory of elasticity. 1 Assuming 
that this problem is solved and that the forces N x , N y , and N xy are known 
at each point of the plate, the components of strain of the middle plane 
of the plate are obtained from the known formulas representing Hooke's 



1 See, for example, S. Timoshenko and J. N. Goodier, " Theory of Elasticity/' 2d 
ed., p. 11, 1951. 



PLATES UNDER LATERAL LOADS AND FORCES IN THEIR PLANE 383 

law, viz., 



£ * = hF ( Nx ~ vN ^ e " = hF (Ar " ~ vNx) 



7x» = 



N 
hG 



(a) 



XV 



The strain energy, due to stretching of the middle plane of the plate, is 
then 

Vi = iIi(N x e x + Ny€ y + N X!/ y X!/ ) dx dy 



= 2^; J J Wl + N l ~ 2vN x N y + 2(1 + v)Nl y ) dx dy (220) 

where the integration is extended over the entire plate. 

Let us now apply the lateral load. This load will bend the plate and 
produce additional strain of the middle plane. In our previous discus- 
sion of bending of plates, this latter strain was always neglected. Here, 

0.14 



0.12 



0.10 



0.08 



t 
& 



0.06 



0.04 



0.02 

















-\ 


'fl/^ 


















^^ 






■£ 






^ 










^s^ 





























0_ 

b 
Fig. 194 



384 



THEORY OF PLATES AND SHELLS 



0.06 



0.05 



0.04 



t 

A 0.03 



0.02 



















s 




^0 
























32- — 












— 7*5—" 





















0_ 

b" 
Fig. 195 



aM*"1b 



however, we have to take it into consideration, since this small strain in 
combination with the finite forces N x , N y , N xy may add to the expression 
for strain energy some terms of the same order as the strain energy of 
bending. The x, y, and z components of the small displacement that a 
point in the middle plane of the plate experiences during bending will be 

denoted by u, v. t and w y respectively. 
Considering a linear element AB of that 
plane in the x direction, it may be seen from 
Fig. 196 that the elongation of the element 
due to the displacement u is equal to 
(du/dx) dx. The elongation of the same 
element due to the displacement w is 
%(dw/dx) 2 dx, as may be seen from the com- 
parison of the length of the element A\B\ in Fig. 196 with the length of 
its projection on the x axis. Thus the total unit elongation in the x direc- 
tion of an element taken in the middle plane of the plate is 



♦uH 



^ 



e 'f "to dx 



•-IS*' 



Fig. 196 



du , 1 /dw 
dx 2 \ta 



)' 



(221) 



PLATES UNDER LATERAL LOADS AND FORCES IN THEIR PLANE 385 



Similarly the strain in the y direction is 



c ' = 



cto 1 /dw 

foj + 2\dy 



y 



(222) 



Considering now the shearing strain in the middle plane due to bend- 
ing, we conclude as before (see Fig. 23) that the shearing strain due to 
the displacements u and v is du/dy + dv/dx. To determine the shear- 
ing strain due to the displacement w we take two infinitely small linear 
elements OA and OB in the x and y directions, as shown in Fig. 197. 
Because of displacements in the z direction these elements come to the 
positions OiAi and 0\B\. The difference between the angle tt/2 and the 
angle AiO\B\ is the shearing strain corresponding to the displacement w. 
To determine this difference we con- 
sider the right angle J5 2 0i^i, in which 
B 2 0i is parallel to BO. Rotating the 
plane B2O1A1 about the axis O1A1 by 
the angle dw/dy, we bring the plane 
BiO\A\ into coincidence with the 
plane J5i0i A 1* and the point B 2 to posi- 
tion C. The displacement B 2 C is equal 
to (dw/dy) dy and is inclined to the ver- 
tical B 2 Bi by the angle dw/dx. Hence 
BiC is equal to (dw/dx) (dw/dy) dy, 
and the angle COiB ly which repre- 
sents the shearing strain corresponding to the displacement w, 
(dw/dx) (dw/dy). Adding this shearing strain to the strain produced by 
the displacements u and v, we obtain 




Fig. 197 



is 



, du dv dw dw 

7xy ~~ &y + dx ~dx dy 



(223) 



Formulas (221), (222), and (223) represent the components of the addi- 
tional strain in the middle plane of the plate due to small deflections. 
Considering them as very small in comparison with the components e x , e y , 
and y xy used in the derivation of expression (220), we can assume that 
the forces N XJ N y , N xy remain unchanged during bending. With this 
assumption the additional strain energy of the plate, due to the strain 
produced in the middle plane by bending, is 

7 2 = S!(N x e' x + N y e' v + N xy y xv ) dx dy 

Substituting expressions (221), (222), and (223) for e£, e' y , and y' xy , we 

* The angles dw/dy and dw/dx correspond to small deflections of the plate and are 
regarded as small quantities. 



386 THEORY OF PLATES AND SHELLS 

finally obtain 



-//[*-£+».5+*-(5+e)]** 

i//[ w -te)'+'-te) ,+ »-£a** (224) 



+ 2 



It can be shown, by integration by parts, that the first integral on the 
right-hand side of expression (224) is equal to the work done during bend- 
ing by the forces acting in the middle plane of the plate. Taking, for 
example, a rectangular plate with the coordinate axes directed, as shown 
in Fig. 192, we obtain for the first term of the integral 

I! jl n - % dx d « = £ M * - £ £ u '£ * * 

Proceeding in the same manner with the other terms of the first integral 
in expression (224), we finally find 

= / ( \N*u\ + \N X J )dy+ I \N y v\ + \N xy u\ ) dx 

Jo \\ 10 I 10/ Jo \\ 10 I 10/ 

-nx^ + ^)-*-n'-(^ + ^)** 

The first integral on the right-hand side of this expression is evidently 
equal to the work done during bending by the forces applied at the edges 
x = and x = a of the plate. Similarly, the second integral is equal to 
the work done by the forces applied at the edges y = and y = b. The 
last two integrals, by virtue of Eqs. (218), are equal to the work done 
during bending by the body forces acting in the middle plane. These 
integrals each vanish in the absence of such corresponding forces. 

Adding expressions (220) and (224) to the energy of bending [see Eq. 
(117), page 88], we obtain the total strain energy of a bent plate under 
the combined action of lateral loads and forces acting in the middle plane 
of the plate. This strain energy is equal to the work T v done by the 
lateral load during bending of the plate plus the work T h done by the 
forces acting in the middle plane of the plate. Observing that this latter 
work is equal to the strain energy Vi plus the strain energy represented 
by the first integral of expression (224), we conclude that the work pro- 



PLATES UNDER LATERAL LOADS AND FORCES IN THEIR PLANE 387 

uced by the lateral forces is 

+!//K0+SO , -*-'>[SS-te) , ]|** 

(225) 

applying the principle of virtual displacement, we now give a variation 8w 
o the deflection w and obtain, from Eq. (225), 

(226) 



+ 



'he left-hand side in this equation represents the work done during the 
irtual displacement by the lateral load, and the right-hand side is the 
orresponding change in the strain energy of the plate. The application 
f this equation will be illustrated by several examples in the next article. 

93. Simply Supported Rectangular Plates under the Combined Action 
f Lateral Loads and of Forces in the Middle Plane of the Plate. Let us 
egin with the case of a rectangular plate uniformly stretched in the 

direction (Fig. 192) and carrying a concentrated load P at a point with 
oordinates £ and tj. The general expression for the deflection that satis- 
es the boundary conditions is 



XV* . rrnrx . rnry , N 

> a mn sm —7- sin - ir {a) 



a b 

m= 1,2,3, .. '. 7i=l,2,3, 



^o obtain the coefficients a mn in this series we use the general equation 
226). Since N y = N xy = in our case, the first integral on the right- 
iand side of Eq. (225), after substitution of series (a) for w, is 

m = 1 n — 1 

^he strain energy of bending representing the second integral in Eq. 
225) is [see Eq. (d), page 343] 

771= 1 71= 1 



388 THEORY OF PLATES AND SHELLS 

To obtain a virtual deflection bw we give to a coefficient a mini an increase 
Sa mini . The corresponding deflection of the plate is 

. miTrx . riiicy 
bw = ba mini sin sin — j— 

The work done during this virtual displacement by the lateral load P is 

P8a mini sin —^ sm -y-' (d) 

The corresponding change in the strain energy consists of the two terms 
which are 

00 00 

ab >r d / V V 9 mV 2 \ . ab Xr rafr 2 , . 

" t ^ 6^; ( 2/ 2/ ^ ~^~ j ,ni = T ,ni "^ 1B1 

m= 1 n= 1 

and 5v = oa mini = — j- Va mini I "^2" + 52 i 4fl «i»i 

Substituting expressions (d) and (e) in Eq. (226), we obtain 



_„ . Wi7r£ . Uiirri ab Ar mfir 2 

P5a mini sm sin — r- = -j- N x a mini —j- oa mini 



from which 



Va 2 "*" 67 



"T ^ J-sQ'm 1 n l 1 ^2 ' k2 I "^ m i n i 



4P sin ^^ sin *£* 

6 (/) 



Substituting these values of the coefficients a miWl in expression (a), we 
find the deflection of the plate to be 

00 00 . miri; . nirri 

A ^ X~A X~^ S111 S111 "T" 

4P \ A \ A a b . mirx . rncy , N 

W = t Ar , 7 7 "7 \9 77- sin sin -TT" W) 



a&7r 4 i) 



/v /v /m^ nA 2 m 2 iV g a 

m-l n-1 \a 2 + 6 2 / + 7T 2 a 2 D 



If, instead of the tensile forces N x , there are compressive forces of the 
same magnitude, the deflection of the plate is obtained by substituting 
— N x in place of N x in expression (g). This substitution gives 

mirx . nwy 
aE^D / j /^/m 2 . n 2 V m% — ~^~ sln 5 

r 2 a 2 Z> 



, D \—\ V~A sin sin IT 

w = 7 JTN > > -t— — rrr- sin sin -T- 2 - (h) 



^2/ 2/ («• + $* 

m = l n = l \a 2 & 2 / 



PLATES UNDER LATERAL LOADS AND FORCES IN THEIR PLANE 389 

The smallest value of N x at which the denominator of one of the terms 
in expression (h) becomes equal to zero is the critical value of the com- 
pressive force N x . It is evident that this critical value is obtained by 
taking n = 1. Hence 

(jV ^ = "Si?- V^ + 5*J = "55" vr + ^y (227) 

where m must be chosen so as to make expression (227) a minimum. 
Plotting the factor 

k 



( mb a V 
\ a mb) 



against the ratio a/b, for various integral values of m, we obtain a system 
of curves shown in Fig. 198. The portions of the curves that must be 



10 















m=1 2 
1 1 


\ X 


L 5 








\ \ 




\ 
\ 








\ '• 


\ \ 


\ 








V 


\ X 


K X 


S- 
S 

-^ 




' 


\ 


^- 


^ -X 


■^"^ 


^ 


s 
^ 




1 






r I - "" 


""""■ I 
1 






1 
1 








1 
I 






1 


1 




1 






1 

1 


1 

1 




1 

I 





1 V2 



2 V6 3 Viz 4 V20 5 





Fig. 198 

used in determining k are indicated by heavy lines. It is seen that the 
factor k is equal to 4 for a square plate as well as for any plate that can 
be subdivided into an integral number of squares with the side 6. It can 
also be seen that for long plates k remains practically constant at a value 
of 4.* Since the value of m in Eq. (227) may be other than 1 for oblong 
plates, such plates, being submitted to a lateral load combined with com- 
pression, do not generally deflect 1 in the form of a half wave in the direc- 
tion of the longer side of the plate. If, for instance, a/b = 2, 4, . • . 
the respective elastic surface becomes markedly unsymmetrical with 
respect to the middle line x = a/2 (Fig. 192), especially so for values of 
N x close to the critical value (N x ) C r- 

By using the deflection (g) produced by one concentrated load, the 

* A more detailed discussion of this problem is given in S. Timoshenko, " Theory of 
Elastic Stability/' p. 327, 1936. 

1 Several examples of such a deformation have been considered by K. Girkmann, 
Stahlbau, vol. 15, p. 57, 1942. 



390 THEORY OF PLATES AND SHELLS 

deflection produced by any lateral load can be obtained by superposition. 

Assuming, for example, that the plate is uniformly loaded by a load of 

intensity q, we substitute q d£ drj for P in expression (g) and integrate 

the expression over the entire area of the plate. In this way we obtain 

the same expression for the deflection of the plate under uniform load as 

has already been derived in another manner (see page 381). 

If the plate laterally loaded by the force P is compressed in the middle 

plane by uniformly distributed forces N x and N y , proceeding as before 

we obtain 

oo * . rrnri . nirn 
a e> V"\ \T"\ sin sin -y- 

on _ 4i ^ \ \ a v • rrnrx . tiny , .. 

w "" „h 4n 7 7 ~7 — ~« \~i sin sin -=-£ (%) 



m=l n = l y a 2 + tfj 



' y 



7r 2 a 2 D tt 2 6 2 D 

The critical value of the forces N x and N y is obtained from the 
condition 1 

m\N x ) cr n*(N y ) cr /m 2 n*\« 
7r 2 a 2 2> """ 7T 2 6 2 Z> \^a 2 "*" 6 2 / U; 

where m and n are chosen so as to make N x and N y a minimum for any 
given value of the ratio N x /N y . In the case of a square plate submitted 
to the action of a uniform pressure p in the middle plane we have a = b 
and N x = N y = p. Equation (j) then gives 

7T 2 D 

Per = — T (m 2 + n 2 ) min (fc) 

The critical value of p is obtained by taking m = n = 1, which gives 

2 ?r 2 D /nnm 

P- = -^- (228) 

In the case of a plate in the form of an isosceles right triangle with 
simply supported edges (Fig. 161) the deflection surface of the buckled 
plate which satisfies all the boundary conditions is 2 



w 



( . ttx . 2iry . 2tx . iru\ 

= a ( sin — sin — - + sin sm — I 

\ a a a a) 



Thus the critical value of the compressive stress is obtained by substi- 
tuting m = 1, n = 2 or m = 2, n = 1 into expression (k). This gives 

Per = -ji- [(229) 

1 A complete discussion of this problem is given in Timoshenko, " Elastic Stability," 
p. 333. 

2 This is the form of natural vibration of a square plate having a diagonal as a nodal 
line. 



PLATES UNDER LATERAL LOADS AND FORCES IN THEIR PLANE 391 



94. Circular Plates under Combined Action of Lateral Load and Tension or Com- 
pression. Consider a circular plate (Fig. 199) submitted to the simultaneous action 
of a symmetrical lateral load and a uniform compression N r = Nt = N in the middle 
plane of the plate. Owing to the slope y? of the deformed plate (Fig. 27) the radial 
compression N gives a transverse component N d<p/dr which we have to add to the 
shearing force Q (Fig. 28) due to the lateral load. 
Hence the differential equation (54) becomes 



dr 2 
in which 



+ 



1 d<p 
r dr 



u( k * l \ Q ^ 



A; 2 = 



Na* 
~D 



GO 



In the case of a circular plate without a hole 1 the 
solution of Eq. (a) is of the form 



tp = C\J i ( — ) 



+ <Po 



(c) 




where Ji is the Bessel function of the order one, <po a particular solution of Eq. (a) 
depending on Q, and C\ a constant defined by the boundary conditions of the plate. 
Let us take as an example a rigidly clamped 2 plate carrying a uniform load of 
intensity q. Then, as a particular solution, we use 



<po 



qra z 
2kW 



qr 

2N 



and therefore 



n - dw - n t 

<p — — — LiJi 
dr 



It follows, by integration, that 



w 



(9-i 



qr 

2N 



qr' 

4N 



(d) 



(e) 



where J is the Bessel function of the order zero and Co a second constant. Having 
calculated C\ from the condition <p = on r = a, and C 2 from the condition w = on 
r = a, we obtain the final solution 3 



qa 4 



Jo \a) ~~ J ° (k) 



qa 2 (a 2 - r 2 ) 



2k*J 1 (k)D 4h*D KJ 

The deflections (/) become infinite for Ji(k) = 0. Denoting the zeros of the func- 
tion Ji in order of their magnitude by j h j i} . . . we see that the condition k = ji 

1 In the case of a concentric hole a term proportional to a Bessel function of second 
kind must be added to expression (c). The inner boundary must be submitted then 
to the same compression N, or else the problem becomes more complex because of the 
inconstancy of stresses N r and N t . 

2 The case of an elastic restraint without transverse load has been discussed by 
H. Reismann, J. Appl. Mechanics, vol. 19, p. 167, 1952. 

3 This result may be found in A. Nddai, "Elastische Flatten," p. 255, Berlin, 1925. 



392 THEORY OF PLATES AND {SHELLS 

defines the lowest critical value 



w„-2S u 

a 2 
of the compressive stress AT. Now, for the function Ji(k) we have the expression 

in which ji - 3.83171, j 2 = 7.01559, .... As fc < ji we can neglect the terms 
k*/j* beginning with the second parentheses. Observing, furthermore, that 

*! . — 

by virtue of Eqs. (b) and (g) we have, approximately, 

J t (k) - ^ (1 - «) (0 

where a = — 0") 

</V C r 

Making use of the expression (i), it can be shown that, approximately, 1 

w q 



w = 



1 - 



where w q is the deflection due to the load q alone. Cases with other boundary condi- 
tions and other laws of distribution of the lateral load may be handled in like manner. 
In the general case of a symmetrical lateral load combined with compression we can 
put, approximately, for the center of the plate (r = 0) 



(w g )o 

wq = 

1 — a 



(Jc) 



0) 



/l dw\ _ /d 2 w\ = 1 +Coa /l dw g \ 
\r llr)o ~ \dr 2 /o ~ 1 - a \r dr Jo 

and on the boundary (r « a) 

A *?\ - 1 + c ' a A dw «\ 
\r dr/a I — * \r dr ) a 

/d*w\ _ 1 + c"* /d 2 w q \ 
\dr*)a " 1 -« \dr 2 /« 

where w ff relates to a plate carrying the given lateral load alone and a ** N/N er has 
the following meaning: 

ATa 2 
For a simply supported plate: a = n 

AMV (m) 

ATa 2 
For a clamped plate: « = ^ 



i 



1 See O. Pettersson, Acta Polytech., Stockholm, no. 138, 1954. The following results 
are taken from this paper, in which, more generally, an elastic restraint at the edge is 
assumed. 



PLATES UNDER LATERAL LOADS AND FORCES IN THEIR PLANE 393 

Table 8i. Values of Constants in Approximate Expressions (k) and (I) 

v = 0.3 



Case 


Load distribution 


Boundary- 
conditions 


Constants 


1 


Uniform edge couples 


Simply supported 


Co = 0.305 
c' = -0.270 
c" = -1.219 


2 


Uniform load 


Simply supported 


Co = 0.0480 

c' = c" = -0.0327 


3 


Clamped 


Co = 0.308 
c" . -0.473 


4 


Central uniform load over area 
of radius ea 


Simply supported 


c ~ 1 1 2153 




1 ' 1 -1.3 In « 
c' = c " = 0.205 


5 


Clamped 


1.308 

Co = — 1 

In e 

c" = 0.0539 



the former value being valid for v = 0.3. The values of the constants c , c', and c" are 
given in Table 81. 

If the circular plate is subjected to a lateral load combined with a uniform tension N, 
instead of compression, then we have, approximately, 



(W)r-O = 



(Wg)r-0 

1 + a 



where a is the absolute value of N/N cr . As for the curvatures, a factor 



1 + (1 + c)a 



instead of the factor (1 + c<x)/(l —a) must be used in expressions (k) and (Z), the 
constant c having the meaning of Co, c', and c", respectively. 

95. Bending of Plates with a Small Initial Curvature. 1 Assume that a 

plate has some initial warp of the middle surface so that at any point 

there is an initial deflection w which is small in comparison with the 

thickness of the plate. If such a plate is submitted to the action of 

transverse loading, additional deflection Wx will be produced, and the 

total deflection at any point of the middle surface of the plate will be 

Wo + wi. In calculating the deflection Wi we use Eq. (103) derived for 

flat plates. This procedure is justifiable if the initial deflection w is 

1 See S. Timoshenko's paper in Mem. InsL Ways Cominun., vol. 89, St. Petersburg, 
1915 (Russian). 



394 THEORY OF PLATES AND SHELLS 

small, since we may consider the initial deflection as produced by a 
fictitious load and apply the principle of superposition. 1 If in addition 
to lateral loads there are forces acting in the middle plane of the plate, 
the effect of these forces on bending depends not only on W\ but also on wq. 
To take this into account, in applying Eq. (217) we use the total deflection 
w = wq + Wi on the right-hand side of the equation. It will be remem- 
bered that the left-hand side of the same equation was obtained from 
expressions for the bending moments in the plate. Since these moments 
depend not on the total curvature but only on the change in curvature 
of the plate, the deflection w 1 should be used instead of w in applying 
that side of the equation to this problem. Hence, for the case of an 
initially curved plate, Eq. (217) becomes 

d 4 wi , 9 d 4 W! dHih JL_ T N d 2 (w + Wi) 

dx* + dx 2 dy 2 + dy 4 D L 9 dx 2 

+ N v d ' {w ' d + ""' + 2^ /' ( 2t y "" ) ] (230) 

It is seen that the effect of an initial curvature on the deflection is equiva- 
lent to the effect of a fictitious lateral load of an intensity 



dx 2 ' y dy 2 ' xy dxdy 

Thus a plate will experience bending under the action of forces in the 
xy plane alone provided there is an initial curvature. 

Take as an example the case of a rectangular plate (Fig. 192), and 
assume that the initial deflection of the plate is defined by the equation 

. 7rx . iry , . 

wq = an sin — sm -r- (a) 

a b 

If uniformly distributed compressive forces N x are acting on the edges 
of this plate, Eq. (230) becomes 

d A Wi . rt d 4 Wi , d 4 Wi 1 /, 7 duir 2 . ttx . iry AT d 2 Wi\ ,,. 
lW +2 *c^ + W = D\ N *-^ Sm ^ Sm T- N --6*) (6) 

Let us take the solution of this equation in the form 

W\ = A sin — sin -^ (c) 

a o 

Substituting this value of W\ into Eq. (6), we obtain 



A = 






1 In the case of large deflections the magnitude of the deflection is no longer pro- 
portional to the load, and the principle of superposition is not applicable. 



PLATES UNDER LATERAL LOADS AND FORCES IN THEIR PLANE 395 

With this value of A expression (c) gives the deflection of the plate pro- 
luced by the compressive forces N x . Adding this deflection to the initial 
deflection (a), Ave obtain for the total deflection of the plate the following 

Bxpression : 

. an . 7r.r . Try ,^ 

w = wo + W! = -z sin — sin -r- (a) 

1 — a a o 

in which 

N x 



a = 



a 2 V ^ &y 

The maximum deflection will be at the center and will be 



(e) 



_ an 

wmax 



1 - a 



(/) 



This formula is analogous to that used for a bar with initial curvature. 1 
In a more general case we can take the initial deflection surface of the 
rectangular plate in the form of the following series : 

00 00 

XV* . mirx . n-Kij , N 

> a mn sin sin -y- (g) 

m = 1 ri = 1 

Substituting this series in Eq. (230), we find that the additional deflection 
at any point of the plate is 



XV* 7 . mirx . nicy /, N 

> b mn sin sin ~r- (h) 

rn = 1 n = 1 



in which 



t*mn^' x / *\ 



tt 2 D ( , n 2 a 2 V A7 
a 2 \ m b 2 / 



It is seen that all the coefficients b mn increase with an increase of JV*. 
Thus when N x approaches the critical value, the term in series (h) that 
corresponds to the laterally buckled shape of the plate [see Eq. (227)] 
becomes the predominating one. We have here a complete analogy with 
the case of bending of initially curved bars under compression. 

The problem can be handled in the same manner if, instead of com- 
pression, we have tension in the middle plane of the plate. In such a 
case it is necessary only to change the sign of N x in the previous equa- 
tions. Without any difficulty we can also obtain the deflection in the 
case when there are not only forces N x but also forces N y and N xy uni- 
formly distributed along the edges of the plate. 

1 See S. Timoshcnko, "Strength of Materials," part II, 3d ed., p. 56, 1956. 



CHAPTER 13 

LARGE DEFLECTIONS OF PLATES 



96. Bending of Circular Plates by Moments Uniformly Distributed 
along the Edge. In the previous discussion of pure bending of circular 
plates it was shown (see page 47) that the strain of the middle plane of 
the plate can be neglected in cases in which the deflections are small as 
compared with the thickness of the plate. In cases in which the deflec- 
tions are no longer small in comparison with the thickness of the plate 
but are still small as compared with the other dimensions, the analysis of 
the problem must be extended to include the strain of the middle plane 
of the plate. 1 

We shall assume that a circular plate is bent by moments Mo uni- 
formly distributed along the edge of the plate (Fig. 200a). Since the 
deflection surface in such a case is symmetrical with respect to the center 
0, the displacement of a point in the middle plane of the plate can be 
resolved into two components: a component u in the radial direction and 
a component w perpendicular to the plane of the plate. Proceeding as 
previously indicated in Fig. 196 (page 384), we conclude that the strain 
in the radial direction is 2 

du , 1 /dw\ 2 , x 

The strain in the tangential direction is evidently 

€ < = - (&) 

Denoting the corresponding tensile forces per unit length by N r and 

1 This problem has been discussed by S. Timoshenko; see Mem. TnsL Ways 
Commun.f vol. 89, St. Petersburg, 1915. 

2 In the case of very large deflections we have 



~ dr + 2 l\dr) + \ dr) J 



which modifies the following differential equations. See E. Reissner, Proc. Symposia 
Appl. Math., vol. 1, p. 213, 1949. 

396 



LARGE DEFLECTIONS OF PLATES 

N t and applying Hooke's law, we obtain 
Eh 



397 



;V, = 



(e r + ve t ) 



= _Eh^\du ,l(dwV Ml 

1 - v*[dr ^ 2\drJ r \ 

Eh , . N Eh \u . du . v/dwVl 



(c) 



These forces must be taken into consideration in deriving equations of 
equilibrium for an element of the plate such as that shown in Fig. 2006 




Mr*f r dr 



N r + dF dr 



dQr 



and c. Taking the sum of the projections in the radial direction of all 
the forces acting on the element, we obtain 



dN r 
dr 



from which 



dr dd + N r dr dO - N t dr dd = 



N '- N - + rW = 



(d) 



398 THEORY OF PLATES AND SHELLS 

The second equation of equilibrium of the element is obtained by taking 
moments of all the forces with respect to an axis perpendicular to the 
radius in the same manner as in the derivation of Eq. (55) (page 53). 
In this way we obtain 1 

n — _ n ( & %w a. * ^ w __ * dw \ ( \ 

The magnitude of the shearing force Q r is obtained by considering the 
equilibrium of the inner circular portion of the plate of radius r (Fig. 
200a). Such a consideration gives the relation 

Substituting this expression for shearing force in Eq. (e) and using expres- 
sions (c) for N r and N t we can represent the equations of equilibrium (d) 
and (e) in the following form: 

d 2 u ___ldtt,w_l — v fdw\ 2 dw d 2 w 



/dw\ 2 __ dw 
\dr J dr 



dr 2 r dr r 2 2r \dr J dr dr 2 

d z w 1 d 2 w 1 ' Jrt " 1 °^» r ^' -• i /^«\2i \^di) 

dr z r dr 2 ~~ 9 



1 dw , 12 dw Tdt6 t* 1 /dwVl 



These two nonlinear equations can be integrated numerically by start- 
ing from the center of the plate and advancing by small increments in 
the radial direction. For a circular element of a small radius c at the 
center, we assume a certain radial strain 



- (— \ 
\dr/ r3SS Q 



and a certain uniform curvature 



I = _ ( dho \ 

po \dr 2 J rsss Q 



With these values of radial strain and curvature at the center, the values 
of the radial displacement u and the slope dw/dr f or r = c can be calcu- 
lated. Thus all the quantities on the right-hand side of Eqs. (231) are 
known, and the values of d 2 u/dr 2 and of d z w/dr z f or r = c can be calcu- 
lated. As soon as these values are known, another radial step of length c 
can be made, and all the quantities entering in the right-hand side of 
Eqs. (231) can be calculated for r = 2c* and so on. The numerical 

1 The direction for Q r is opposite to that used in Fig. 28. This explains the minus 
sign in Eq. (e). 

* If the intervals into which the radius is divided are sufficiently small, a simple 
procedure, such as that used in S. Timoshenko's " Vibration Problems in Engineering," 
3d ed., p. 143, can be applied. The numerical results represented in Fig. 201 are 



LARGE DEFLECTIONS OF PLATES 



399 



values of u and w and their derivatives at the end of any interval being 
known, the values of the forces N r and N t can then be calculated from 
Eqs. (c) and the bending moments M r and M t from Eqs. (52) and (53) 
(see page 52). By such repeated calculations we proceed up to the radial 
distance r = a at which the radial force N r vanishes. In this way we 
obtain a circular plate of radius a bent by moments M Q uniformly dis- 
tributed along the edge. By changing the numerical values of € and 



14 


±t&s 


6Mr/h^: 
















n 












































( 


>M t / 


fi*' 












12 






















































occne E ^ 
For stresses :1 division = . 9 10 

For deflection M division = 0.05 h 




















10 


































w 0J 




/\N 






8 




































6 


















































































































































4 




































































































2 






































N r /h, 


























































■r- 





















































































N t /h- 














-2 
































1* 


1 


I 











10 
h 

Fig. 201 



15 



20 



1 /p at the center we obtain plates with various values of the outer radius 
and various values of the moment along the edge. 

Figure 201 shows graphically the results obtained for a plate with 



a 



2Sh and (M r )r=a = M = 2.93 • 10" 3 



D 

h 



It will be noted that the maximum deflection of the plate is 0.55/i, which 
is about 9 per cent less than the deflection w given by the elementary 
theory which neglects the strain in the middle plane of the plate. The 
forces N r and N t are both positive in the central portion of the plate. 
In the outer portion of the plate the forces N t become negative; i.e., 



obtained in this manner. A higher accuracy can be obtained by using the methods 
of Adams or Stormer. For an account of the Adams method see Francis Bashforth's 
book on forms of fluid drops, Cambridge University Press, 1883. Stormer's method 
is discussed in detail in A. N. Krilov's book " Approximate Calculations/ ' pub- 
lished by the Russian Academy of Sciences, Moscow, 1935. See also L. Collatz, 
"Numerische Behandlung von Differentialgleichungen, ,, Berlin, 1951. 



400 THEORY OF PLATES AND SHELLS 

compression exists in the tangential direction. The maximum tangential 
compressive stress at the edge amounts to about 18 per cent of the maxi- 
mum bending stress QMo/h 2 . The bending stresses produced by the 
moments M r and M t are somewhat smaller than the stress 6M /h 2 given 
by the elementary theory and become smallest at the center, at which 
point the error of the elementary theory amounts to about 12 per cent. 
From this numerical example it may be concluded that for deflections of 
the order of 0.5h the errors in maximum deflection and maximum stress 
as given by the elementary theory become considerable and that the 
strain of the middle plane must be taken into account to obtain more 
accurate results. 

97, Approximate Formulas for Uniformly Loaded Circular Plates with 
Large Deflections. The method used in the preceding article can also be 
applied in the case of lateral loading of a plate. It is not, however, of 
practical use, since a considerable amount of numerical calculation is 
required to obtain the deflections and stresses in each particular case. 
A more useful formula for an approximate calculation of the deflections 
can be obtained by applying the energy method. 1 Let a circular plate 
of radius a be clamped at the edge and be subject to a uniformly dis- 
tributed load of intensity q. Assuming that the shape of the deflected 
surface can be represented by the same equation as in the case of small 
deflections, we take 

/ r 2\2 

w = Wo M - -\ (a) 

The corresponding strain energy of bending from Eq. (w) (page 345) is 

7== 2J Jolw) + A^J + T^^\ rdrd = ~TT* D 

For the radial displacements we take the expression 

u = r {a - r)(Ci + C 2 r + C z r* + • • •) (c) 

each term of which satisfies the boundary conditions that u must vanish 
at the center and at the edge of the plate. From expressions (a) and (c) 
for the displacements, we calculate the strain components € r and u of the 
middle plane as shown in the preceding article and obtain the strain 
energy due to stretching of the middle plane by using the expression 



f, - 2. /; («? + ^s) r * - ^ /; W + 4 + *** * 



(d) 



1 See Timoshenko, "Vibration Problems," p. 452. For approximate formulas see 
also Table 82. 



LARGE DEFLECTIONS OF PLATES 401 

Taking only the first two terms in series (c), we obtain 

y. = TEha \ (o.250C?a 2 4- 0.1167C!a 4 + 0.300CiC 2 a 3 
1 — v 2 \ 

- 0.00846Caa ^ + 0.00682C 2 a 2 ^° + 0.00477 ^) (e) 

The constants Ci and C 2 are now determined from the condition that the 

total energy of the plate for a position of equilibrium is a minimum. 

Hence 

dV i n a dV > n (ft 

Wi = and ^ = (/) 

Substituting expression (e) for V ly we obtain two linear equations for Ci 
and C%. From these we find that 

d = 1.185^2 and C, = -1.75^ 
a 3 a 4 

Then, from Eq. (e) we obtain 1 

V 1 = 2.59x1) -^ (?) 

Adding this energy, which results from stretching of the middle plane, 
to the energy of bending (&), we obtain the total strain energy 

V + V,-|rD^(l +0.2«ff) W 

The second term in the parentheses represents the correction due to strain 
in the middle surface of the plate. It is readily seen that this correction 
is small and can be neglected if the deflection w Q at the center of the plate 
is small in comparison with the thickness h of the plate. 

The strain energy being known from expression (h), the deflection of 
the plate is obtained by applying the principle of virtual displacements. 
From this principle it follows that 

d(V + Vj) 6wq = ^ r qSwrdr = 2irq 8w Q / ° ( 1 - ^ J r dr 
dw Jo Jo \ a / 

Substituting expression (h) in this equation, we obtain a cubic equation 
for Wq. This equation can be put in the form 

u, = -g£ — i s (232) 

UD 1 + 0.488 ^ 

The last factor on the right-hand side represents the effect of the stretch- 
ing of the middle surface on the deflection. Because of this effect the 
deflection Wq is no longer proportional to the intensity q of the load, and 

1 It is assumed that v = 0.3 in this calculation. 



402 THEORY OF PLATES AND SHELLS 

the rigidity of the plate increases with the deflection. For example, 
taking wq = %h, we obtain, from Eq. (232), 

w Q = 0.89 qa * 



64D 



This indicates that the deflection in this case is 1 1 per cent less than that 
obtained by neglecting the stretching of the middle surface. 

Up to now we have assumed the radial displacements to be zero on the 
periphery of the plate. Another alternative is to assume the edge as free 
to move in the radial direction. The expression (232) then has to be 
replaced by 

w ° = ms 1 — *« (233) 

^ U 1 +0.146^2 

a result 1 which shows that under the latter assumption the effect of the 
stretching of the plate is considerably less marked than under the former 
one. Taking, for instance, Wo = %h we arrive at Wq = 0.965(ga 4 /64Z>), 
with an effect of stretching of only 3£ per cent in place of 11 per cent 
obtained above. 

Furthermore we can conclude from Eqs. (&) and (c) of Art. 96 that, if 
N r = on the edge, then the edge value of N t becomes N t = Ehe t = Ehu/r, 
that is, negative. We can expect, therefore, that for a certain critical 
value of the lateral load the edge zone of the plate will become unstable. 2 

Another method for the approximate solution of the problem has been 
developed by A. Nddai. 3 He begins with equations of equilibrium simi- 
lar to Eqs. (231). To derive them we have only to change Eq. (/), of the 
preceding article, to fit the case of lateral load of intensity q. After such a 
change the expression for the shearing force evidently becomes 



dr 



Using this expression in the same manner in which expression (/) was 
used in the preceding article, we obtain the following system of equations 
in place of Eqs. (231): 

d 2 u 1 du _ u __ __ 



dr 2 r dr r 



2 



d z w 1 d 2 w 
dr z r dr 2 



1 - 

2r 


H 


'du* 
K drj 


I*" 


dw d 2 w 
dr dr 2 


dw f 


du 
dr 


T 


+ ii 


/dw 
[dr 



(234) 



1 Obtained by a method which will be described in Art. 100. 

2 The instability occurring in such a case has been investigated by D. Y. Panov and 
V. I. Feodossiev, Priklad. Mat. Mekhan., vol. 12, p. 389, 1948. 

3 See his book "Elastische Flatten/' p. 288, 1925. 



LARGE DEFLECTIONS OF PLATES 403 

To obtain an approximate solution of the problem a suitable expression 
for the deflection w should be taken as a first approximation. Substi- 
tuting it in the right-hand side of the first of the equations (234), we 
obtain a linear equation for u which can be integrated to give a first 
approximation for u. Substituting the first approximations for u and w 
in the right-hand side of the second of the equations (234), we obtain a 
linear differential equation for w which can be integrated to give a second 
approximation for w. This second approximation can then be used to 
obtain further approximations for u and w by repeating the same sequence 

of calculations. 

In discussing bending of a uniformly loaded circular plate Avith a 
clamped edge, Nadai begins with the derivative dw/dr and takes as first 
approximation the expression 



dw _ c [" r _ /A" 
~dr ~~ [ a \ a / 



U) 



which vanishes f or r = and r = a in compliance with the condition at 
the built-in edge. The first of the equations (234) then gives the first 
approximation for u. Substituting these first approximations for u and 
dw/dr in the second of the equations (234) and solving it for q, we deter- 
mine the constants C and n in expression (J) so as to make q as nearly a 
constant as possible. In this manner the following equation 1 for calcu- 
lating the deflection at the center is obtained when v = 0.25: 

£ + 0.583 (?)" -0.176 | (J)' (235) 

In the case of very thin plates the deflection w may become very large 
in comparison with h. In such cases the resistance of the plate to bend- 
ing can be neglected, and it can be treated as a flexible membrane. The 
general equations for such a membrane are obtained from Eqs. (234) by 
putting zero in place of the left-hand side of the second of the equations. 
An approximate solution of the resulting equations is obtained by neg- 
lecting the first term on the left-hand side of Eq. (235) as being small in 
comparison with the second term. Hence 

0.583 feY - 0.176 § (|) 4 and w = 0.665a 

1 Another method for the approximate solution of Eqs. (234) was developed by 
K. Federhofer, Eisenbau, vol. 9, p. 152, 1918; see also Forschungsarb. VDI, vol. 7, 
p. 148, 1936. His equation for w« differs from Eq. (235) only by the numerical value 
of the coefficient on the left-hand side; viz., 0.523 must be used instead of 0.583 for 
v = 0.25. 




404 THEORY OF PLATES AND SHELLS 

A more complete investigation of the same problem 1 gives 

31 




(236) 



This formula, which is in very satisfactory agreement with experiments, 2 
shows that the deflections are not proportional to the intensity of the load 
but vary as the cube root of that intensity. For the tensile stresses at 
the center of the membrane and at the boundary the same solution gives, 
respectively, 



(cr r ) r „o = 0.423 



^ 



EqW 
h 2 



and (<r r )r-a = 0.328 



3 /ffg 2 q 2 



To obtain deflections that are proportional to the pressure, as is often 
required in various measuring instruments, recourse should be had to 
corrugated membranes 3 such as that shown in Fig. 202. As a result of 

the corrugations the deformation con- 
sists primarily in bending and thus 
increases in proportion to the pressure. 4 
If the corrugation (Fig. 202) follows a 
sinusoidal law and the number of 
waves along a diameter is sufficiently large (n > 5) then, with the nota- 
tion of Fig. 186, the following expression 5 for wq = (w) m&x may be used: 




Fig. 202 



8 



(?)[ 



3(1 - » 2 ) 



+ 



(ffhm-i& 



98. Exact Solution for a Uniformly Loaded Circular Plate with a 
Clamped Edge. 6 To obtain a more satisfactory solution of the problem 
of large deflections of a uniformly loaded circular plate with a clamped 
edge, it is necessary to solve Eqs. (234). To do this we first write the 
equations in a somewhat different form. As may be seen from its deri- 

1 The solution of this problem was given by H. Hencky, Z. Math. Physik, vol. 63, 
p. 311, 1915. For some peculiar effects arising at the edge zone of very thin plates 
see K. O. Friedrichs, Proc. Symposia Appl. Math., vol. 1, p. 188, 1949. 

2 See Bruno Eck, Z. angew. Math. Mech., vol. 7, p. 498, 1927. For tests on circular 
plates with clamped edges, see also A. McPherson, W. Ramberg, and S. Levy, NACA 
Rept. 744, 1942. 

3 The theory of deflection of such membranes is discussed by K. Stange, Ingr.-Arch., 
vol. 2, p. 47, 1931. 

4 For a bibliography on diaphragms used in measuring instruments see M. D. 
Hersey's paper in NACA Rept. 165, 1923. 

* A. S. Volmir, "Flexible Plates and Shells,' ' p. 214, Moscow, 1956. This book alsa 
contains a comprehensive bibliography on large deflections of plates and shells. 
6 This solution is due to S. Way, Trans. ASME, vol. 56, p. 627, 1934. 



LARGE DEFLECTIONS OF PLATES 405 

vation in Art. 96, the first of these equations is equivalent to the equation 

N r - N t + r^ = (237) 

Also, as is seen from Eq. (e) of Art. 96 and Eq. (i) of Art. 97, the second 
of the same equations can be put in the following form: 

~ /d 3 w . 1 d 2 w 1 dw\ Ar dw qr mR v 

D \d? + -rd? ~ 7* Tr) = N 'dF + 2 (238) 

From the general expressions for the radial and tangential strain (page 
396) we obtain 

, de t , 1 fdw\ 

er = et + r d7 + 2\-d7) 

Substituting 

er = i {Nr " vNt) and €t = ^E (Nt " ^ r) 
in this equation and using Eq. (237), we obtain 

r £ V,, + Nt +%(%)' -0 (239) 

The three Eqs. (237), (238), and (239) containing the three unknown 
functions N r , N t , and w will now be used in solving the problem. We 
begin by transforming these equations to a dimensionless form by intro- 
ducing the following notations: 

With this notation, Eqs. (237), (238), and (239) become, respectively, 

j- (iSr) -S t = (241) 



12(1 - ^)dH^^\ dr /J 2 ' dr 

tl^+w+Ky)'- (243) 

The boundary conditions in this case require that the radial displace- 
ment u and the slope dw/dr vanish at the boundary. Using Eq. (6) of 
Art. 96 for the displacements u and applying Hooke's law, these con- 
ditions become 

(U)r-a = r(S t - vS r )r=a = (244) 

(a) 



(y)~ - ° 



406 THEORY OF PLATES AND SHELLS 

Assuming that S r is a symmetrical function and dw/dr an antisym- 
metrical function of £, we represent these functions by the following 
power series: 

& = Bo + B 2 ? + B A ? + • • • (6) 

^ = VS (Ci{ + C 3 £ 3 + C.{» + • • •) (c) 

in which Bo, jB 2 , . . . and Ci, C 3 , . . . are constants to be determined 
later. Substituting the first of these series in Eq. (241), we find 

St = Bo + ZB t ? + bB& + • • • (d) 

By integrating and differentiating Eq. (c), we obtain, respectively, 



J- ^(cf + cf + *$+-..) 

(^) = V8 (C a + 3C 3 £ 2 + 5C 5 * 4 + • • 



<«) 



^(^) = V8 (C a + 3Cs£ 2 + 5C 5 * 4 + • • •) (/) 

It is seen that all the quantities in which we are interested can be found 
if we know the constants Bo, B 2 , . . . , Ci, C 3 , . . . . Substituting 
series (6), (c), and (d) in Eqs. (242) and (243) and observing that these 
equations must be satisfied for any value of £, we find the following 
relations between the constants B and C: 

k-i 

±>k = TTT i ~7y\ / lsmlsk—m /C = J, 4, O, • • • 





4 


y 


k(k + 2) 

1 

12(1 - p 2 ) 


L, 

n= 1,3,5, 

i 

= 2 4 


/c 2 - 


- 1 

m 



^* = 7,2 Z 1 /j BmCk-2-m k = 5, 7, 9, . . . (gf) 

m = 0,2,4,... 

Cs - 1 (1 - - !) (wi + B ' c ) 

It can be seen that when the two constants B and C\ are assigned, all the 
other constants are determined by relations (g). The quantities S r , S t , 
and dw/dr are then determined by series (6), (d), and (c) for all points in 
the plate. As may be seen from series (6) and (/), fixing B and d is 
equivalent to selecting the values of S r and the curvature at the center 
of the plate. 1 

To obtain the following curves for calculating deflections and stresses in 
particular cases, the procedure used was: For given values of v and 

1 The selection of these same quantities has already been encountered in the case of 
bending of circular plates by moments uniformly distributed along the edge (see 
page 398). 



LARGE DEFLECTIONS OF PLATES 



407 



p = q/E and for selected values of B and C h a considerable number of 
numerical cases were calculated, 1 and the radii of the plates were deter- 
mined so as to satisfy the boundary condition (a). For all these plates 
the values of S r and S t at the boundary were calculated, and the values of 
the radial displacements (w) r=a at the boundary were determined. Since 
all calculations were made with arbitrarily assumed values of B and Ci, 
the boundary condition (244) was not satisfied. However, by interpo- 
lation it was possible to obtain all the necessary data for plates for which 
both conditions (244) and (a) are satisfied. The results of these calcu- 
lations are represented graphically in Fig. 203. If the deflection of the 

2.0 



1.6 






£ 1-2 



c 
o 



sS 0.8 
















v=0.25^ 


*N. 




















v=0L30». 


•^. 














Elementary theory 


^X« 












































































\*< 


















































^v = 0.3S 





























































< 


I 


i 




( 


L« 


I 


J 


1 





i: 


i 



0.4 



Load, £^4 
Fig. 203 

plate is found from this figure, the corresponding stress can be obtained 
by using the curves of Fig. 204. In this figure, curves are given for the 
membrane stresses 

Nr 

h 
and for the bending stresses 

6M r 



(T r = 



0". = 



/l 2 



as calculated for the center and for the edge of the plate. 2 By adding 
together <s T and o>, the total maximum stress at the center and at the 
edge of the plate can be obtained. For purposes of comparison Figs. 
203 and 204 also include straight lines showing the results obtained from 

1 Nineteen particular cases have been calculated by Way, op. cit. 

2 The stresses are given in dimensionless form. 



408 



THEORY OF PLATES AND SHELLS 



the elementary theory in which the strain of the middle plane is neg- 
lected. It will be noted that the errors of the elementary theory increase 
as the load and deflections increase. 



7 




i i i i 

Bending stress 


f 










at edge* * 1 










6 






















Bending stress by- 
linear theory, edge 


/-. 










5 

C4 |cM_ 






















Bending stress 
. by //near theory 


*«» / 


* ^ 










b| uj 
S 4 


center 








' *». 




























t/3 

3 














/Bending stress 












a 


i at center 




2 












l__ KA~ 








ss— 












^ Mem or an e srre 
i at center 

1 L 1 I 


1 










/ 


^i Membrane sh 


•ess 


1 












/j 


^ at edge 




























a- 


] 


0.1 
Def 


3 

lecti 
204 


Li 
on,V 


k/t 


u 




L8 



99. A Simply Supported Circular Plate under Uniform Load. An exact solution of 
the problem 1 can be obtained by a series method similar to that used in the preceding 
article. 

Because of the axial symmetry we have again dw/dr = and N r = Nt at r *= 0. 
Since the radial couples must vanish on the edge, a further condition is 



[d /dw\ v dw~\ _ 
dr\drj r dr J r -a 



(a) 



With regard to the stress and strain in the middle plane of the plate'two boundary 
conditions may be considered: 

1. Assuming the edge is immovable we have, by Eq. (244), St — vS r = 0, which, 
by Eq. (237), is equivalent to 



[*tt-.>+rfL-0 



(&) 



1 K. Federhofer and H. Egger, Sitzber. Akad. Wiss. Wien, Ha, vol. 155, p. 15, 1946; 
see also M. Stippes and A. H. Hausrath, J. Appl. Mechanics, vol. 19, p. 287, 1952. 
The perturbation method used in this latter paper appears applicable in the case of a 
concentrated load as well. 



LARGE DEFLECTIONS OF PLATES 409 

2. Supposing the edge as free to move in the radial direction we simply have 

GS r )r-. = (c) 

The functions S r and dw/dr may be represented again in form of the series 

S ' = io/i k2 t^ « (BlP + B * pZ + Bbpb + ' ' '> (d) 

12(1 — *> 2 )a 2 p 

^ ^ (C 1P + C 3P 3 + C hP * + • • (e) 

<*r 2a V 3 

where p = r/a. Using these series and also Eqs. (241), (242), (243), from which the 
quantity S r can readily be eliminated, we arrive at the following relations between 
the constants B and C: 

k-2 



(/) 



Bk = " 2(fc 2 - 1) Li CmCk ~ m - 1 * - 3i 5, • - - 

m= 1,3,5, ... 
Jfc-2 

C* = 77^-t / C m B k - m -x k = 5, 7, . . . (ff) 

A; 2 - 1 Z-/ 

m = 1,3,5, . . . 

8C 3 - B 1 C 1 + 12 V3 (1 - p 2 ) ^ = (ft) 

where p = g/2?, g being the intensity of the load. 

Again, all constants can easily be expressed in terms of both constants Bi and Ci, 
for which two additional relations, ensuing from the boundary conditions, hold: 

In case 1 we have 

Y B k (k - = J C k (k + 0=0 (t) 

Jfc= 1,3,5, ... fc = 1,3,5, ... 

and in case 2 

T B fc = y C k (k + 0=0 (j) 

ib« 1,3,5,... Jb-1,3,5,... 

To start the resolution of the foregoing system of equations, suitable values of B Y and 
Ci may be taken on the basis of an approximate solution. Such a solution, satisfying 
condition (a), can be, for instance, of the form 

dw „, /f v 

— = C(p P » - p) (k) 

dr 

where CAs a constant and p = (n = 3, 5, . . .). Substituting this in Eqs. 

n + v 

(241) and (243), in which £ must be replaced by pa/h, and eliminating & we obtain 

c 2 C 2 / p 2n p n+1 p 2 \ 

p 2 2 \ rti n 2 8 / 



410 



THEORY OF PLATES AND SHELLS 




3 5 10 25 50. 100 200 308 Eh 4 

Logarithmic scale forobscissas 

Fig. 205 



Table 82. Data for Calculation of Approximate Values of Deflections 

wo and Stresses in Uniformly Loaded Plates 

v - 0.3 



Boundary 


A 


B 


Center 


Edge 


conditions 


oc T = at 


/Sr = /3 t 


OCT 


oct 


0r 


fit 


Plate 
clamped 


Edge im- 
movable 


0.471 


0.171 


0.976 


2.86 


0.476 


0.143 


-4.40 


-1.32 




Edge free 
to move 


0.146 


0.171 


0.500 


2.86 





-0.333 


-4.40 


-1.32 


Plate 
simply 
supported 


Edge im- 
movable 


1.852 


0.696 


0.905 


1.778 


0.610 


0.183 





0.755 


Edge free 
to move 


0.262 


0.696 


0.295 


1.778 





-0.427 





0.755 



LARGE DEFLECTIONS OF PLATES 411 

Herein cy and ci are constants of integration and 

rii = 4n(n + 1) 7i2 = (n + l)(n + 3) 
Let us, for example, assume the boundary conditions of case 2. Then we obtain 



Cl ~ "2 \ni " n^ + 8/ 



c 2 = 



(m) 



The constant C, finally, can be determined by some strain energy method — for exam- 
ple, that described in Art. 100. Using there Eqs. (?/*) or (o) we have only to replace 

80 






70 



60 



10r 50 



40 



Rodi 


ally 


1 

movab 


1 

le edge- 


1 

i/ = 


.25 




j 


~(0tV = 
































/ 


J 
















/ 




w /h 














1/ 






















(<r)r»o 




































qo 4 



- 30 

(Up 

h 

- 20 



- 10 



oL o_„ _____ ______ 

1 3 5 10 25 50 100 200 308 Eh 4 

Logarithmic scale for abscissas 

Fig. 206 

dip/dr = rhESr and dw/dr by approximate expressions in accordance with Eqs. (&) 
and (Z) given above. 

The largest values of deflections and of total stresses obtained by Federhofer and 
Kgger from the exact solution are given in Fig. 205 for case 1 and in Fig. 206 for case 2. 
The calculation has been carried out for v = 0.25. 

Table 82 may be useful for approximate calculations of the deflection wo at the 



412 THEORY OF PLATES AND SHELLS 

center, given by an equation of the form 



Wo 

T 



+ '(?)*-*i(0' « 



also of the stresses in the middle plane, given by 



<r r ass a r E — - <r< = a*2J — (o) 

a 2 a 2 

and of the extreme fiber bending stresses 1 

/ „ Woh / ^ _ woh 

Q? Or 

100. Circular Plates Loaded at the Center. An approximate solution of this 
problem can be obtained by means of the method described in Art. 81. 

The work of the internal forces corresponding to some variation de rj 8e t of the strain is 

bVi = -2ir J ° (N r 5e r + N t 8e t )r dr 
Using Eqs. (a) and (6) of Art. 96 we have 

We assume, furthermore, that either the radial displacements in the middle plane or 
the radiai forces N r vanish on the boundary. Then, integrating expression (a) by 
parts and putting bu « or N r = on r = a, we obtain 

f a T d 1 f a dw /dw\ 

The work of the bending moments M r and M t on the variation 8(—d 2 w/dr 2 ) and 
$(— i dw/dr) of the curvatures is similarly 

•*-»-£MSW'(£)]'* 

Now we suppose that either the radial bending moment M r or the slope S(dw/dr) 
becomes zero on the boundary. Integration of expression (c) by parts then yields 

$y 2 = 2*r / ° D L ( A w)5 (^ J r dr (d) 

Finally, the work of the external forces is 

6V Z =2x1 q 8wrdr 



or, by putting 



i f r , 

i/r ts — I qr dr 

r Jo 



1 d , x , . 

q - ~ -r (nW (e) 

r ar 



1 The sign is negative if the bottom of the plate is in compression. 



LARGE DEFLECTIONS OF PLATES 413 

we have 



Wo" 



. a d 
hV z = 2tt / — (rt) bw r dr 



dr 
Provided bw = on the boundary we finally obtain 



fa /dw\ 
SV* - -2tt / r$b ( — J r dr (/) 



W 



The condition b(Vi + V 2 + V z ) = now yields the equation 

f a \dj(Aw) - * - iV r ^ 1 ^ (to) r dr + j \j r {rN T ) - N t bu dr = 

We could proceed next by assuming both variations bw and bu as arbitrary. Thus 
we would arrive at the second of the differential equations (234), N r being given by 
expression (c) of Art. 96, and at Eq. (d) of the same article. If we suppose only this 
latter equation of equilibrium to be satisfied, then we have still to fulfill the condition 



f a \ d s 1 dfdwl d , x T 

/ D — (Aw) - ^ 7 -tIt (M r dr = 

/ o dr r dr dr J dr 



(h) 



in which / is a stress function defining 



r or ar* 



and governed by the differential equation 



_ _Ehfdw\ i 
2r \dr/ 



jWJ --==( = ) Cfl 



which follows from Eq. (239). Integrating expression (h) by parts once more we 
obtain 

/ Z>AAw - q -[-f—)\bwrdr=0 (k) 

/o L r d r \dr dr J J 

With intent to use the method described in Art. 81 we take the deflection in the form 

w = ai<pi(r) + a 2 <P2(r) + • • • + a n <p n (r) (I) 

Just as in the case of the expression (211) each function <pi(r) has to satisfy two 
boundary conditions prescribed for the deflection. Substituting expression (I) either 
in Eq. (h) or in Eq. (A;) and applying the same reasoning as in Art. 81, we arrive at a 
sequence of equations of the form 



/. 



° X^rrdr i - 1,2, . . . ,n (m) 

dr 



in which 



^ d , 1 dfdw , . 

X - D — (Aw) -+---+— (n) 

dr r dr dr 



414 THEORY OF PLATES AND SHELLS 

or at a set of equations 

/ Yipir dr i = 1, 2, . . . , n (o) 

1 d /dfdw\ 
where Y = DAAw - q J \7 IT ) &) 

Now let us consider a clamped circular plate with a load P concentrated at r = 0. 
We reduce expression (/) to its first term by taking the deflection in the form 



w 



/ r 2 r 2 A 

-u>ol 1 -- +2- log-) 
\ « a a / 



(?) 



which holds rigorously for a plate with small deflections. From Eq. (j) we obtain, 
by integration, 



df Ehwlr* / r 3, r , 7\ , ^ C 

dr a 4 \ a 2 a 8/ r 



(r) 
r 



Let there be a free radial displacement at the boundary. The constants of integra- 
tion Ci and C 2 then are determined by two conditions. The first, namely, 

(N r )r=a = 

can be rewritten as 



es~-° 



(•) 
\r ar/r-a 

and the second is 

(0 



\dr/r-o 



This latter condition must be added in order to limit, at r = 0, the value of the 
stress N r given by Eq. (i). Thus we obtain 



Cl -I^ C 2 =0 
8 a 2 



The load function is equal to 



P 

r 27rr 



in our case, and expressions (q) and (r) yield 



v 8w P , 4Eii>Jfc /r 3 , r 3 r 3 r 7 r 3 r 7 r A 

X = D — I — log 3 log 2 - + - — log - log - J (?/) 

ah 2wr a 3 \a 3 * a 2 a 3 * a 8 a 3 * a 8a a/ 

while ^i is given by the expression in the parentheses in Eq. (q). Substituting this in 
Eq. (m) we arrive at the relation 

191 Pa 2 

16 Dwo + — Ehwl (v) 

648 7r 



LARGE DEFLECTIONS OF PLATES 415 

The general expressions for the extreme fiber bending stresses corresponding to the 
deflection (q) and obtainable by means of Eqs. (101.) are 



2Ehwo \ a 

- — (1 + log - - 1 

(1 — v 2 )a 2 I r 



lEhwo r 
- v*)a* |_ ' 

• r 



2Ehw „ , v , a 
°t = t; 7T\ (1 + log v 

1 (1 - ? 2 )a 2 r 



(w) 



These expressions yield infinite values of stresses as r tends to zero. However, assum- 
ing the load P to be distributed uniformly over a circular area with a small radius 
r = c, we can use a simple relation existing in plates with small deflections between 
the stresses a" = a" at the center of such an area and the stresses c r = <r' t caused at 
r = c by the same load P acting at the point r = 0. According to N&dai's result, 1 
expressed in terms of stresses, 

// // / . & P 

Applying this relation to the plate with large deflections we obtain, at the center of 
the loaded area with a radius c, approximately 



)a 2 L 



„ 2Ehw \ a 

*t = ~ — I (1 + ^) log - ~ 1 



(1 - *> 2 )a 2 I c 



3 p 

H 0*0 

2 tt/i 2 



The foregoing results hold for a circular plate with a clamped and movable edge. 
By introducing other boundary conditions we obtain for w Q an equation 

(2/) 






which is a generalization of Eq. (v). The constants A and B are given in Table 83. 
The same table contains several coefficients 2 needed for calculation of stresses 

<r r = ot r rj — a t = c^/i — \Z) 

a 2 a 2 

acting in the middle plane of the plate and the extreme fiber bending stresses 

/ „ woh , „ woh , , x 

*' T -PrE— *' t =p,E— («') 

a 2 a 1 

The former are calculated using expressions (z), the latter by means of expressions 
(101) for the moments, the sign being negative if the compression is at the bottom. 3 

101. General Equations for Large Deflections of Plates. In discussing 
the general case of large deflections of plates we use Eq. (219), which was 

1 A. Nddai, "Elastische Platten," p. 63, Berlin, 1925. 

2 All data contained in Table 82 are taken from A. S. Volmir, op. cit. 

3 For bending of the ring-shaped plates with large deflections see K. Federhofer, 
Osterr. Ingr.-Arch., vol. 1, p. 21, 1946; E. Reissner, Quart. Appl. Math., vol. 10, p. 167, 
1952, and vol. 11, p. 473, 1953. Large deflections of elliptical plates have been dis- 
cussed by N. A. Weil and N. M. Newmark, /. Appl Mechanics, vol. 23, p. 21, 1956. 



416 



THEORY OH 1 PLATES AND SHELLS 



Table 83. Data for Calculation of Approximate Values of Deflections 
w* and Stresses in Centrally Loaded Plates 

v - 0.3 



Boundary- 


A 


B 


Center 


Edge 


conditions 


otr = at 


Otr 


at 


fir 


fit 


Plate 
clamped 


Edge im- 
movable 


0.443 


0.217 


1.232 


0.357 


0.107 


-2.198 


-0.659 




Edge free 
to move 


0.200 


0.217 


0.875 





-0.250 


-2.198 


-0.659 


Plate 
simply 
supported 


Edge im- 
movable 


1.430 


0.552 


0.895 


0.488 


0.147 





0.606 


Edge free 
to move 


0.272 


0.552 


0.407 





-0.341 





0.606 



derived by considering the equilibrium of an element of the plate in the 
direction perpendicular to the plate. The forces N X) N v , and N xy now 
depend not only on the external forces applied in the xy plane but also 
on the strain of the middle plane of the plate due to bending. Assuming 
that there are no body forces in the xy plane and that the load is perpen- 
dicular to the plate, the equations of equilibrium of an element in the 
xy plane are 

dN X dNyy = Q 

dV (a) 



dx 
dN> 



dx 



*V I dJS y _ q 



dy 



VI 



The third equation necessary to determine the three quantities N X) AT, 
and N xy is obtained from a consideration of the strain in the middle sur- 
face of the plate during bending. The corresponding strain components 
[see Eqs. (221), (222), and (223)] are 



e* = 



Yxy — 



dx + 2\dx) 

dv.l /dvA 2 
dy + 2\dy) 

du . dv . dw dw 
~dy d~x d~xd~y 



Q>) 



By taking the second derivatives of these expressions and combining the 



LARGE DEFLECTIONS OF PLATES 417 

resulting expressions, it can be shown that 

a^/ 2 + dx 2 dx dy \dx dy) dx 2 dy 2 ^ 

By replacing the strain components by the equivalent expressions 

€x = hE ^ N * ~ vNy) 

€ » = -KE {Nv ~ vNx) (d) 



7xv ~ hG N 



xy 



the third equation in terms of N Xf N y , and N xy is obtained. 

The solution of these three equations is greatly simplified by the intro- 
duction of a stress function. 1 It may be seen that Eqs. (a) are identically 
satisfied by taking 



d 2 F d 2 F d 2 F 

W N v = h w N ** = ^ h Wd~y 



■XT t ° r HT T. ° r -KT -L ° F / \ 



where F is a function of x and y. If these expressions for the forces are 
substituted in Eqs. (d), the strain components become 



~ E\dy 2 V dx 2 ) 



1 /d 2 F d 2 F\ 
2(1 + v) d 2 F 



E dx dy 
Substituting these expressions in Eq. (c), we obtain 



d A F d 4 F dW 

dx A + dx 2 dy 2 + dy 



7 rrlY^wY d 2 wd 2 wl ,_,_,. 

* = E [(dx-Ty) ~dx- 2 W\ (245) 



The second equation necessary to determine F and w is obtained by 
substituting expressions (e) in Eq. (217), which gives 

d*w . „ d*w , d*w h (q , d 2 F d 2 w 



)x* + dx 2 dv 2 dv 4 DU dv 2 c 



dx* ' dx 2 dy 2 ' ay 4 D \h ' dy 2 dx 2 

a 2 ffa 2 w a 2 F a 2 w \ , . 

"*" dx 2 dy 2 dxdydxdy) K ' 

1 See S. Timoshenko and J. N. Goodier, "Theory of Elasticity," 2d ed., p. 26. 1951. 



418 THEORY OF PLATES AND SHELLS 

Equations (245) and (246), together with the boundary conditions, 
determine the two functions F and w.* Having the stress function F, 
we can determine the stresses in the middle surface of a plate by apply- 
ing Eqs. (e). From the function w, which defines the deflection surface 
of the plate, the bending and the shearing stresses can be obtained by 
using the same formulas as in the case of plates with small deflection [see 
Eqs. (101) and (102)]. Thus the investigation of large deflections of 
plates reduces to the solution of the two nonlinear differential equations 
(245) and (246). The solution of these equations in the general case is 
unknown. Some approximate solutions of the problem are known, how- 
ever, and will be discussed in the next article. 

In the particular case of bending of a plate to a cylindrical surface 1 
whose axis is parallel to the y axis, Eqs. (245) and (246) are simplified by 
observing that in this case w is a function of x only and that d 2 F/dx 2 and 
d 2 F/dy 2 are constants. Equation (245) is then satisfied identically, and 
Eq. (246) reduces to 

d 4 w _ jf, N x d 2 w 
dx A ~~ D + D dx 2 

Problems of this kind have already been discussed fully in Chap. 1. 

If polar coordinates, more convenient in the case of circular plates, are used, the 
system of equations (245) and (246) assumes the form 

E 
AAF = — -- L(w,w) 

AAw - - L(w,F) + Q - 



in which 

d 2 w 



(1 dF 1 d 2 F \ (idw 1 dhv\ d*F 9 d_ / 1 dF\ j9 / 1 dw\ 
r&r r 2 dd 2 ) \r dr r 2 dd 2 J dr 2 ~~ " dr \r dd J dr \r 36 ) 



and L(v)jW) is obtained from the foregoing expression if w is substituted for F. 

In the case of very thin plates, which may have deflections many times 
larger than their thickness, the resistance of the plate to bending can be 

* These two equations were derived by Th. von KdrmAn; see " Encyklopadie der 
Mathematischen Wissenschaften," vol. IV 4 , p. 349, 1910. A general method of non- 
linear elasticity has been applied to bending of plates by E. Koppe, Z. angew. Math. 
Mech., vol. 36, p. 455, 1956. 

1 For a more general theory of plates (in particular of cantilever plates) bent, with- 
out extension, to a developable surface, see E. H. Mansfield, Quart. J. Mech. AppL 
Math., vol. 8, p. 338, 1955, and D. G. Ashwell, Quart. J. Mech. AppL Math., vol. 10, 
p. 169, 1957. A boundary-layer phenomenon arising along the free edges of such 
plates was considered by Y. C. Fung and W. H. Witrick, Quart. J. Mech. AppL Math., 
vol. 8, p. 191, 1955. 



LARGE DEFLECTIONS OF PLATES 419 

neglected; i.e., the flexural rigidity D can be taken equal to zero, and 
the problem reduced to that of finding the deflection of a flexible mem- 
brane. Equations (245) and (246) then become 1 



'bx 4 + bx 2 by 2 + by 4 ~ | \bx by) bx 2 by 2 } 
h by 2 bx 2 + bx 2 by 2 bx by bx by 



(247) 



A numerical solution of this system of equations by the use of finite 
differences has been discussed by H. Hencky. 2 

The energy method affords another means of obtaining an approxi- 
mate solution for the deflection of a membrane. The strain energy of a 
membrane, which is due solely to stretching of its middle surface, is given 
by the expression 

V = ifj(N x e x + NyCy + N xu y xu ) dx dy 
Eh 



2(i -„*)// [el + € * + 2v€ *» + i0 " v)yl " ] dx dy (248) 

Substituting expressions (221), (222), and (223) for the strain compo- 
nents e x , e y , y xy , we obtain 



V= Eh 



MduY du (dwY (dv\ dv (dwY 
dx) + dx \dx) + \dy) + dy \dy) 



2(1 

2 l\dy) dy dx + \dx) + dy dx dy 



dv dw dw 

~* dx dx dy 



dx dy (249) 



In applying the energy method we must assume in each particular case 
suitable expressions for the displacements u, v, and w. These expressions 
must, of course, satisfy the boundary conditions and will contain several 
arbitrary parameters the magnitudes of which have to be determined by 
the use of the principle of virtual displacements. To illustrate the 
method, let us consider a uniformly loaded square membrane 3 with sides 
of length 2a (Fig. 207). The displacements u, v, and w in this case must 
vanish at the boundary. Moreover, from symmetry, it can be concluded 

1 These equations were obtained by A. Foppl, "Vorlesungen uber Technische 
Mechanik," vol. 5, p. 132, 1907. 

2 H. Hencky, Z. angew. Math. Mech., vol. 1, pp. 81 and 423, 1921; see also R. Kaiser, 
Z. angew. Math. Mech., vol. 16, p. 73, 1936. 

3 Calculations for this case are given in the book "Drang und Zwang" by August 
and Ludwig Foppl, vol. 1, p. 226, 1924; see also Hencky, ibid. 



420 



THEORY OF PLATES AND SHELLS 



that w is an even function of x and y, whereas u and v are odd functions 

of x and of y, respectively. All these requirements 
are satisfied by taking the following expressions for 
the displacements: 




TTX TV 

w = Wo cos 7r - cos -pr 
2a 2a 

. TTX wy 

u = c sin — cos pr- 

a 2a 

. iry ttx 
v = c sin — COS t^- 
CI 2a 



^ 



which contain two parameters wo and c. Substituting these expressions 
in Eq. (249), we obtain, for v = 0.25, 



7.5 [ 64 a 2 



17ir 2 cu>§ . 2 /35tt 2 . 80 



)] 



■(*) 



The principle of virtual displacements gives the two following equations: 1 

dV 
dc 



= 



dV_ 
dw 



6wt 



/ +a f +a * * x Try j j 

-. J - a Q 8W ° C ° S 2~a COS Ya dx dy 



(i) 
U) 



Substituting expression (h) for V, we obtain from Eq. (i) 



Wo 
a 



and from Eq. (j) 



0.147 



wo = 0.802a Jjj£ 




qa 



(250) 



This deflection at the center is somewhat larger than the value (236) 
previously obtained for a uniformly loaded circular membrane. The 
tensile strain at the center of the membrane as obtained from expressions 
(g) is 



e x = e v = — = 0.462 



a 



a' 



t 



q*Ea? 



and the corresponding tensile stress is 

<r = J®- 0.462 !f| = 0.616^ = 0.396./^: (251) 

1 — v a 1 a 1 \ Ar 

Some application of these results to the investigation of large deflections 
of thin plates will be shown in the next article. 

1 The right-hand side of Eq. (t) is zero, since the variation of the parameter c pro- 
duces only horizontal displacements and the vertical load does not produce work. 



LARGE DEFLECTIONS OF PLATES 421 

102. Large Deflections of Uniformly Loaded Rectangular Plates. We begin with 
the case of a plate with clamped edges. To obtain an approximate solution of the 
problem the energy method will be used. 1 The total strain energy V of the plate is 
obtained by adding to the energy of bending [expression (117), page 88] the energy 
due to strain of the middle surface [expression (249), page 419]. The principle of 
virtual displacements then gives the equation 

dV - Sffqw dx dy = (a) 

which holds for any variation of the displacements u, v, and w. By deriving the vari- 
ation of V we can obtain from Eq. (a) the system of Eqs. (245) and (246), the exact 
solution of which is unknown. To find an approximate solution of our problem we 
assume for u, v, and w three functions satisfying the boundary conditions imposed by 
the clamped edges and containing several parameters which will be determined by 
using Eq. (a). For a rectangular plate with sides 2a and 26 and coordinate axes, as 
shown in Fig. 207, we shall take the displacements in the following form: 

u = (a 2 - x 2 j(b 2 - y 2 )x(b o + 6022/ 2 + b 2 ox 2 + b 22 x 2 y 2 ) 
v = (a 2 - x 2 )(b 2 - y 2 )y(c Q0 + c 02 y 2 + c 20 x 2 + c 22 x 2 y 2 ) (b) 

w = (a 2 - x 2 ) 2 {b 2 - y 2 ) 2 (a o + a 02 y 2 + a 20 x 2 ) 

The first two of these expressions, which represent the displacements u and v in the 
middle plane of the plate, are odd functions in x and y, respectively, and vanish at 
the boundary. The expression for w, which is an even function in x and y f vanishes 
at the boundary, as do also its first derivatives. Thus all the boundary conditions 
imposed by the clamped edges are satisfied. 

Expressions (6) contain 11 parameters 600, . . . , a 2 o, which will now be determined 
from Eq. (a), which must be satisfied for any variation of each of these parameters. 
In such a way we obtain 11 equations, 3 of the form 

£(y- JJ *>**)- ° w 

and 8 equations of the form 2 

dV dV 
— — = or (d) 

uOmn dCmn 

These equations are not linear in the parameters a mn , bmn, and c mn as was true in the 
case of small deflections (see page 344). The three equations of the form (c) will con- 
tain terms of the third degree in the parameters a mn . Equations of the form (d) will 
be linear in the parameters b mn and c mn and quadratic in the parameters a wn . A solu- 
tion is obtained by solving Eqs. (d) for the b mn 's and c mn 's in terms of the a mn J B and 
then substituting these expressions in Eqs. (c). In this way we obtain three equa- 

1 Such a solution has been given by S. Way; see Proc. Fifth Intern. Congr. Appl. 
Mech. y Cambridge, Mass., 1938. For application of a method of successive approxi- 
mation and experimental verification of results see Chien Wei-Zang and Yeh Kai- 
Yuan, Proc. Ninth Intern. Congr. Appl. Mech., Brussels, vol. 6, p. 403, 1957. Large 
deflections cf slightly curved rectangular plates under edge compression were con- 
sidered by Syed Yusuff, /. Appl. Mechanics, vol. 19, p. 446, 1952. 

2 The zeros on the right-hand sides of these equations result from the fact that the 
lateral load does not do work when u or v varies. 



422 



THEOKY OF PLATES AND SHELLS 



tions of the third degree involving the parameters a mn alone. These equations can 
then be solved numerically in each particular case by successive approximations. 

Numerical values of all the parameters have been computed for various intensities 
of the load q and for three different shapes of the plate b/a = 1, b/a =* f , and b/a « £ 
by assuming v = 0.3. 

It can be seen from the expression for w that, if we know the constant a o, we can 
at once obtain the deflection of the plate at the center. These deflections are graphi- 
cally represented in Fig. 208, in which w m *x/h is plotted against qb A /Dh. For com- 
parison the figure also includes the straight lines which represent the deflections 
calculated by using the theory of small deflections. Also included is the curve for 
b/a = 0, which represents deflections of an infinitely long plate calculated as explained 
in Art. 3 (see page 13). It can be seen that the deflections of finite plates with 
b/a < | are very close to those obtained for an infinitely long plate. 

Knowing the displacements as given by expressions (6), we can calculate the strain 
of the middle plane and the corresponding membrane stresses from Eqs. (6) of the 

2.0 



1.0 



a~2^ 


I 


a 3 p. 




* 




*N 




2 




b 


of 


'/ 
















\ 


<b_ 
a" 


1 


a > 




~3 




























S 











































































































































100 



qb 4 
Dh 



200 



Fig. 208 

preceding article. The bending stresses can then be found from Eqs. (101) and (102) 
for the bending and twisting moments. By adding the membrane and the bending 
stresses, we obtain the total stress. The maximum values of this stress are at the 
middle of the long sides of plates. They are given in graphical form in Fig. 209. For 
comparison, the figure also includes straight lines representing the stresses obtained 
by the theory of small deflections and a curve b/a = representing the stresses for 
an infinitely long plate. It would seem reasonable to expect the total stress to be 
greater for b/a = than for b/a = £ for any value of load. We see that the curve 
for b/a = falls below the curves for b/a = i and b/a = |. This is probably a 
result of approximations in the energy solution which arise out of the use of a finite 
number of constants. It indicates that the calculated stresses are in error on the 
safe side, i.e., that they are too large. The error for b/a = £ appears to be about 
10 per cent. 

The energy method can also be applied in the case of large deflections of simply 
supported rectangular plates. However, as may be seen from the foregoing dis- 
cussion of the case of clamped edges, the application of this method requires a con- 
siderable amount of computation. To get an approximate solution for a simply 



LARGE DEFLECTIONS OF PLATES 



423 



supported rectangular plate, a simple method consisting of a combination of the 
known solutions given by the theory of small deflections and the membrane theory 
can be used. 1 This method will now be illustrated by a simple example of a square 
plate. We assume that the load q can be resolved into two parts q\ and q* in such a 
manner that part q\ is balanced by the bending and shearing stresses calculated by 



















































/ 
























/ 








15 












t 




j 


/ 


















~BT2 fc 


/, 


K 




Z 1 


\h 


=1 - 








b. 
a 


i i 


f 




// 


£"■ 


/ 




a 








ii 


1 




V 














CM 

7> 


MO 








1 




// 




















b 


1 


1 


A 




















a' 


'1/ 


V 




/ 






















1, 


/A 


f / 


/ 






















k 


7/ 


/ 






















L 


" , 


'/ 




















5 




1 


7 


f 






















/ 


/ 
























i 


/ 


























I 


/ 


























/ 


























°0 


r 


































10 



it;. 


qb 4 
Dh 

20!) 






2C 


)0 







the theory of small deflections, part q 2 being balanced by the membrane stresses. 
The deflection at the center as calculated for a square plate with sides 2a by the 
theory of small deflections is 2 



From this we determine 



wo = 0.730 



qia 4 
Eh* 



0i = 7T^ 



woEh* 



0.730a 4 



(e) 



1 This method is recommended by Foppl; see " Drang und Zwang/' p. 345. 

2 The factor 0.730 is obtained by multiplying the number 0.00406, given in Table 8. 
by 16 and by 12(1 - v 2 ) = 11.25. 



424 THEORY OF PLATES AND SHELLS 

Considering the plate as a membrane and using formula (250), we obtain 

3/ 

wo = 0.802a x P=r 
V Eh 



from which 

0.516a 4 
The deflection w is now obtained from the equation 



wlEh 

?2 = „ g , g A if) 



WpEh* w\Eh 

g=gi+g2 = O730^ + O516^ 



which gives 



= ^!(:.37 + 1 . 9 4|-;) < 2 52> 



After the deflection wo has been calculated from this equation, the loads qi and q 2 are 
found from Eqs. (e) and (/), and the corresponding stresses are calculated by using 
for qi the small deflection theory (see Art. 30) and for q 2) Eq. (251). The total stress 
is then the sum of the stresses due to the loads q\ and q 2 . 

Another approximate method of practical interest is based on consideration of the 
expression (248) for the strain energy due to the stretching of the middle surface of 
the plate. 1 This expression can be put in the form 



E -hJI 



V - Ek ., II [e* - 2(1 - v)e 2 ] dx dy (g) 

in which 

e = €* + c y e 2 = €*«y — \y\ v 

A similar expression can be written in polar coordinates, e 2 being, in case of axial 
symmetry, equal to c r €«. The energy of bending must be added, of course, to the 
energy (g) in order to obtain the total strain energy of the plate. Yet an examination 
of exact solutions, such as described in Art. 98, leads to the conclusion that terms of 
the differential equations due to the presence of the term e 2 in expression (g) do not 
much influence the final result. 

Starting from the hypothesis that the term containing e 2 actually can be neglected 
in comparison with e*, we arrive at the differential equation of the bent plate 

(h) 

in which the quantity 

19 Tai/ a*! 1 /a*/>\* 1 /ai/>\*T 

(i) 



ar = — I 



i 


&Aw - 


- a 2 Aw = 


9 
D 


dU 

dx 


dy 


+ii 


fdw S 
[dx. 


)' 



proves to be a constant. From Eqs. (6) of Art. 101 it follows that the dilatation 
e = tx 4- €y then also remains constant throughout the middle surface of the bent 
plate. The problem in question, simplified in this way, thus becomes akin to prob- 
lems discussed in Chap. 12. 

1 H. M. Berger, J. Appl. Mechanics, vol. 22, p. 465, 1955. 



LARGE DEFLECTIONS OF PLATES 425 

For a circular plate under symmetrical loading, Eq. (i) must be replaced by 

12 [da u 1 /dw\*l 

In this latter case the constants of integration of JEq. (h) along with the constant a 
allow us to fulfill all conditions prescribed on the boundary of the plate. However, 
for a more accurate calculation of the membrane stresses N r , N t from the deflections, 
the first of the equations (231) should be used in place of the relation (j). 

The calculation of the membrane stresses in rectangular plates proves to be rela- 
tively more cumbersome. As a whole, however, the procedure still remains much 
simpler than the handling of the exact equations (245) and (246), and the numerical 
results, in cases discussed till now, prove to have an accuracy satisfactory for technical 
purposes. Nevertheless some reservation appears opportune in application of this 
method as long as the hypothesis providing its basis lacks a straight mechanical 
interpretation. 

103. Large Deflections of Rectangular Plates with Simply Supported Edges. An 
exact solution 1 of this problem, treated in the previous article approximately, can be 
established by starting from the simultaneous equations (245) and (246). 

The deflection of the plate (Fig. 59) may be taken in the Navier form 



00 00 

-11 

m= 1 n= 1 



. mwx . nwy 
w = 7 7 Wmn sin sin — - (a) 



the boundary conditions with regard to the deflections and the bending moments 
thus being satisfied by any, yet unknown, values of the coefficients w mn - The given 
lateral pressure may be expanded in a double Fourier series 



00 00 

-22 



. mwx . nwy 

q mn sin sin — - (6) 

a b 



m = 1 n = 1 
A suitable expression for the Airy stress function, then, is 



P x y 2 P v x* y V mwx nwy 

F = — — + — — + / / /«n cos cos — (c) 

2bh 2ah Lj Z-/ a b 

m = ?i = 



where P x and P y denote the total tension load applied on the sides x = 0, a and 
y = 0, 6, respectively. Substituting the expressions (a) and (c) into Eq. (245), we 
arrive at the following relation between the coefficients of both series : 



i 



fmn = 77 —. , - /M2 ) , br* Pq WrsW pq (d) 

4(m 2 6/a + n 2 a/b) 2 f s 



1 Due to S. Levy, NACA Tech. Note 846, 1942, and Proc. Symposia Appl. Math., 
vol. 1, p. 197, 1949. For application of the same method to plates with clamped 
edges see this latter paper and NACA Tech. Notes 847 and 852, 1942; for application 
to slightly curved plates under edge compression see J. M. Coan, J. Appl. Mechanics, 
vol. 18, p. 143, 1951. M. Stippes has applied the Ritz method to the case where 
the membrane forces vanish on the boundary and two opposite edges are supported ; 
see Proc. First Natl. Congr. Appl. Mech., Chicago, 1952, p. 339. 



426 THEORY OF PLATES AND SHELLS 

The sum includes all products for which r ± p = m and s ± q — n. The coefficients 
brspq are given by the expression 

brs Pq = 2rspq ± (r 2 g 2 + s 2 p 2 ) (e) 

where the sign is positive for r + p = m and s — q = n or for r — p = w and 
-|- g = n, and is negative otherwise. Taking, for example, a square plate (a = 6), 
we obtain 

E 

/ 2>4 = — (-4ty lfl ty lt3 -f- 36w?i,i^3,3 + 36t0i,it01.6 + 64Wi, 2 Wl,6 • • •) 

1,600 

It still remains to establish a relation between the deflections, the stress function, and 
the lateral loading. Inserting expressions (a), (6), and (c) into Eq. (246), we arrive 
at the equation 



/m 2 n 2 Y n ™*** , r> nV , h * A V n x 

q mn = DWrnn** [^ + ~J + P,W mn ~^ + P>W-» ^ + ^ /^ <Wr.U>, 



(/) 



The summation includes, this time, all products for which r ± p = m and s ± q = n, 
and the coefficients are given by 

Crs Pq = ±(rq ± sp) 2 if r ^ and s ^ (g) 

and are twice this value otherwise. The first sign is positive if either r — p = m or 
s _ q = n (but not simultaneously), and is negative in all other cases. The second 
sign is positive if r + p = m and s - g = n or r - p = in and s + q = n, and is 
negative otherwise. For example, 



/l 9\ 2 



2 Q 2 



for 4 
H — (-8/0,2^1.1 - 8/o,2^1.5 + 100/2,4^3.1 - 64/ 2t 2^3.1 + • * 

4a 2 o 2 

In accordance with conditions occurring in airplane structures the plate is con- 
sidered rigidly framed, all edges thus remaining straight 1 after deformation. Then 
the elongation of the plate, say in the direction x, is independent of y. By Eqs. (6) 
and (/) of Art. 101 its value is equal to 

' Jo dx Jo lE\dy* dx*J 2\3x/J 

Using the series (a) and (c), this yields 



~ bkE hE 8a Z/ Z/ 



m = l n = 1 

i.e., an expression which in fact does not include y. Similarly, one obtains 

00 00 

m=l n = 1 
1 A solution due to Kaiser, loc. ciL, is free from this restriction. 



LARGE DEFLECTIOXS OF PLATES 



427 



12 



£0_ 

Eh 2 





<. — 





> 








l 
1 


A 


B 


A 




I 
1 
1 



1 
1 

1 


C 


D 


C 








1 


A 


e 


A 


/ Membrane stresses 




> 


i 






- y 






immovable edges, 
v =0.316 












Tens^- — 












Tension 





(er x ) D =(<r y ) l , 



100 



go 4 
Eh 4 

Fig. 210 



200 



(cr x ) B =(cr y ) c 



(cr x ) A =(a y } A 



300 




300 



Fig. 211 



428 THEORY OF PLATES AND SHELLS 

With regard to the boundary conditions we again consider two cases: 

1. All edges are immovable. Then h x = 5„ = and Eqs. (t) and (j) allow us to 
express P x and P y through the coefficients w mn - 

2. The external edge load is zero in the plane of the plate. We have then simply 

P X - Py » 0. 

Next we have to keep a limited number of terms in the series (a) and (6) and to 
substitute the corresponding expressions (d) in Eq. (/). Thus we obtain for any 
assumed number of the unknown coefficients w mn as many cubic equations. Having 
resolved these equations we calculate the coefficients (d) and are able to obtain all 
data regarding the stress and strain of the plate from the series (a) and (c). The 
accuracy of the solution can be judged by observing the change in the numerical 
results as the number of the coefficients w mn introduced in the calculation is gradually 
increased. Some data for the flexural and membrane stresses obtained in this manner 
in the case of a uniformly loaded square plate with immovable edges are given in 
Figs. 210 and 211. 



CHAPTER 14 

DEFORMATION OF SHELLS WITHOUT BENDING 



104. Definitions and Notation. In the following discussion of the 
deformations and stresses in shells the system of notation is the same as 
that used in the discussion of plates. We denote the thickness of the 
shell by h, this quantity always being considered small in comparison 
with the other dimensions of the shell and with its radii of curvature. 
The surface that bisects the thickness of the plate is called the middle 
surface. By specifying the form of the middle surface and the thickness 
of the shell at each point, a shell is entirely defined geometrically. 

To analyze the internal forces w T e cut from the shell an infinitely small 
element formed by two pairs of adjacent planes which are normal to the 
middle surface of the shell and which contain its principal curvatures 
(Fig. 212a). We take the coordinate axes x and y tangent at to the 
lines of principal curvature and the axis z normal to the middle surface, 
as shown in the figure. The principal radii of curvature which lie in the 
xz and yz planes are denoted by r x and r y , respectively. The stresses 
acting on the plane faces of the element are resolved in the directions of 
the coordinate axes, and the stress components are denoted by our previ- 
ous symbols a x , a y , r xy = r yx , r X2 . With this notation 1 the resultant forces 
per unit length of the normal sections shown in Fig. 212b are 



(a) 



n - - \Z °* - dz *- = \Z °* ( i - dz 

r+h/2 / z \ r+h/2 / z \ 

N xy = / r xy ( 1 - - )dz N yx = r yx ( 1 - - ) dz 

J-h/2 \ r y ) J- h/2 \ r x ) 

Q x = / ' T**(l - 0r) dz Qy = / ^l 1 ""^)^ ( c ) 

J-h/2 \ r y J J- h/2 \ r x ) 

The small quantities z/r x and z/r y appear in expressions (a), (&), (c), 
because the lateral sides of the element shown in Fig. 212a have a trape- 
zoidal form due to the curvature of the shell. As a result of this, the 
shearing forces N xy and N yx are generally not equal to each other, although 

1 In the cases of surfaces of revolution in which the position of the element is defined 
by the angles 6 and <p (see Fig. 213) the subscripts and <p are used instead of x and y 
in notation for stresses, resultant forces, and resultant moments. 

429 



430 



THEORY OF PLATES AND SHELLS 



it still holds that r xy = r yx . In our further discussion we shall always 
assume that the thickness h is very small in comparison with the radii 
r x , r v and omit the terms z/r x and z/r y in expressions (a), (6), (c). Then 
N xy = N yx , and the resultant shearing forces are given by the same 
expressions as in the case of plates (see Art. 21). 




Fig. 212 



The bending and twisting moments per unit length of the normal sec- 
tions are given by the expressions 



M, 



M 



xy 






CxZ 



+h/2 
h/2 






dz 



M; 



■ - 1: 
-/- 



h/2 
+h/2 

h/2 






dz 



id) 



in which the rule used in determining the directions of the moments is 
the same as in the case of plates. In our further discussion we again 
neglect the small quantities z/r x and z/r y , due to the curvature of the 
shell, and use for the moments the same expressions as in the discussion 

of plates. 

In considering bending of the shell, we assume that linear elements, 
such as AD and BC (Fig. 212a), Avhich are normal to the middle surface 
of the shell, remain straight and become normal to the deformed middle 
surface of the shell. Let us begin with a simple case in which, during 



DEFORMATION OF SHELLS WITHOUT BENDING 43 L 

bending, the lateral faces of the element ABCD rotate only with respect 
to their lines of intersection with the middle surface. If r z and r' y are the 
values of the radii of curvature after deformation, the unit elongations of 
a thin lamina at a distance z from the middle surface (Fig. 212a) are 



""rflfe '*- 1 *_£\r' 1l r y ) 



(/) 



If, in addition to rotation, the lateral sides of the element are displaced 
parallel to themselves, owing to stretching of the middle surface, and if 
the corresponding unit elongations of the middle surface in the z and y 
directions are denoted by ei and e 2 , respectively, the elongation e x of the 
lamina considered above, as seen from Fig. 212c, is 



li — ti 
Substituting 



Ave obtain 



I, = ds (l - f\ h = rfs(l + €i) (l - 

_J 21 

I - *!>', r x \ 



!_£ i_£ 

r x r x 



(9) 



A similar expression can be obtained for the elongation e„. In our fur- 
ther discussion the thickness h of the shell will be always assumed small 
in comparison with the radii of curvature. In such a case the quantities 
z/r x and z/r y can be neglected in comparison with unity. We shall neg- 
lect also the effect of the elongations a and e 2 on the curvature. 1 Then, 
instead of such expressions as (g), we obtain 



«. = <i ~ *{? x ~ 7x ) = «i " ** 



where Xx and x y denote the changes of curvature. Using these expres- 
sions for the components of strain of a lamina and assuming that there 
are no normal stresses between laminae (a z = 0), the following expres- 

1 Similar simplifications are usually made in the theory of bending of thin curved 
bars. It can be shown in this case that the procedure is justifiable if the depth of the 
cross section h is small in comparison with the radius r, say h/r < 0.1; see S. Timo- 
shenko, "Strength of Materials/' part I, 3d ed., p. 370, 1955. 



432 THEORY OF PLATES AND SHELLS 

sions for the components of stress are obtained: 

E 



1 - v 2 
E 



Gv = 



„2 



[ei + vez - z(xx + vxy)] 
[€ 2 + vei - z(xv + vXx)] 



Substituting these expressions in Eqs. (a) and (d) and neglecting the 
small quantities z/r x and z/r y in comparison with unity, we obtain 

N x = r — ^ (ex + ^ 2 ) N y = j— ^ (e 2 + ve x ) (253) 

M* = -IKx* + vxv) My = -D(xi/ + vxx) 

where D has the same meaning as in the case of plates [see Eq. (3)] and 
denotes the flexural rigidity of the shell. 

A more general case of deformation of the element in Fig. 212 is 
obtained if we assume that, in addition to normal stresses, shearing 
stresses also are acting on the lateral sides of the element. Denoting 
by 7 the shearing strain in the middle surface of the shell and by Xx V dx 
the rotation of the edge BC relative to Oz about the x axis (Fig. 212a) and 
proceeding as in the case of plates [see Eq. (42)], we find 

r*v = (t - 2z X xy)G 

Substituting this in Eqs. (6) and (e) and using our previous simplifications, 
we obtain 

iv xy - iv yx - 2(1 + ^ (2 54) 

M X y = -My X = D(l - V)XX V 

Thus assuming that during bending of a shell the linear elements normal 
to the middle surface remain straight and become normal to the deformed 
middle surface, we can express the resultant forces per unit length N x , 
N y , and N xy and the moments M x , M y , and M xy in terms of six quantities: 
the three components of strain e h e 2 , and y of the middle surface of the 
shell and the three quantities Xx, Xy, &nd Xx V representing the changes of 
curvature and the twist of the middle surface. 

In many problems of deformation of shells the bending stresses can be 
neglected, and only the stresses due to strain in the middle surface of the 
shell need be considered. Take, as an example, a thin spherical container 
submitted to the action of a uniformly distributed internal pressure nor- 
mal to the surface of the shell. Under this action the middle surface of 
the shell undergoes a uniform strain; and since the thickness of the shell 
is small, the tensile stresses can be assumed as uniformly distributed 
across the thickness. A similar example is afforded by a thiu circular 



DEFORMATION OF SHELLS WITHOUT BENDING 



433 



cylindrical container in which a gas or a liquid is compressed by means of 
pistons which move freely along the axis of the cylinder. Under the 
action of a unif crm internal pressure the hoop stresses that are produced 
in the cylindrical shell are uniformly distributed over the thickness of 
the shell. If the ends of the cylinder are built in along the edges, the 
shell is no longer free to expand laterally, and some bending must occur 
near the built-in edges when internal pressure is applied. A more com- 
plete investigation shows, however f 
(see Art. 114), that this bending is of 
a local character and that the portion 
of the shell at some distance from the 
ends continues to remain cylindrical 
and undergoes only strain in the 
middle surface without appreciable 
bending. 

If the conditions of a shell are such 
that bending can be neglected, the 
problem of stress analysis is greatly 
simplified, since the resultant moments 
(d) and (e) and the resultant shearing 
forces (c) vanish. Thus the only un- 
knowns are the three quantities N x , 
N y , and N xy = N ys , which can be de- 
termined from the conditions of equi- 
librium of an element, such as shown in 
Fig. 212. Hence the problem becomes 
statically determinate if all the forces 
acting on the shell are known. The 




forces N x , N y , and N xy obtained in this 



Fig. 213 



manner are sometimes called membrane forces, and the theory of shells 
based on the omission of bending stresses is called membrane theory. The 
application of this theory to various particular cases will be discussed in 
the remainder of this chapter. 

105, Shells in the Form of a Surface of Revolution and Loaded Sym- 
metrically with Respect to Their Axis. Shells that have the form of 
surfaces of revolution find extensive application in various kinds of con- 
tainers, tanks, and domes. A surface of revolution is obtained by rota- 
tion of a plane curve about an axis lying in the plane of the curve. This 
curve is called the meridian, and its plane is a meridian plane. An ele- 
ment of a shell is cut out by two adjacent meridians and two parallel 
circles, as shown in Fig. 213a. The position of a meridian is defined by 
an angle 0, measured from some datum meridian plane; and the position 
of a parallel circle is defined by the angle <p, made by the normal to the 



434 THEORY OF PLATES AND SHELLS 

surface and the axis of rotation. The meridian plane and the plane 
perpendicular to the meridian are the planes of principal curvature at a 
point of a surface of revolution, and the corresponding radii of curvature 
are denoted by r x and r 2 , respectively. The radius of the parallel circle is 
denoted by r so that the length of the sides of the element meeting at 0, 
as shown in the figure, are r x d<p and r dd = r 2 sin <p dO. The surface 
area of the element is then T\T<l sin ip dip dd. 

From the assumed symmetry of loading and deformation it can be 
concluded that there will be no shearing forces acting on the sides of 
the element. The magnitudes of the normal forces per unit length are 
denoted by N? and N e as shown in the figure. The intensity of the 
external load, which acts in the meridian plane, in the case of symmetry 
is resolved in two components Y and Z parallel to the coordinate axes. 
Multiplying these components with the area rir 2 sin ip dip dd, we obtain 
the components of the external load acting on the element. 

In writing the equations of equilibrium of the element, let us begin 
with the forces in the direction of the tangent to the meridian. On the 
upper side of the element the force 

iV> dd = JV> 2 sin ip dO (a) 

is acting. The corresponding force on the lower side of the element is 

(*.+iH (*+£*)« (6) 

From expressions (a) and (6), by neglecting a small quantity of second 
order, we find the resultant in the y direction to be equal to 

N p d(pdd + dN± ro dipdd = %- (N,r ) dip dd (c) 

* dtp dtp dip 

The component of the external force in the same direction is 

Yrxrodipdd (d) 

The forces acting on the lateral sides of the element are equal to iV>i dip 
and have a resultant in the direction of the radius of the parallel circle 
equal to N e ridipdB. The component of this force in the y direction 
(Fig. 2136) is 

— Neri cos ip dip dO (e) 

Summing up the forces (c), (d), and (e), the equation of equilibrium in 
the direction of the tangent to the meridian becomes 

i- (# f^o) - N ^ c ° s * + 7rir ° = ° o 

dip 



DEFORMATION OF SHELLS WITHOUT BENDING 



435 



The second equation of equilibrium is obtained by summing up the 
projections of the forces in the z direction. The forces acting on the 
upper and lower sides of the element have a resultant in the z direction 
equal to 



N^ro dd dip 



(9) 



The forces acting on the lateral sides of the element and having the 
resultant iV>i dip dd in the radial direction of the parallel circle give a 
component in the z direction of the magnitude 



NoT\ sin <p dip dd 



(h) 



The external load acting on the element has in the same direction a 
component 

Zrir dd dip (i) 

Summing up the forces (g), (h), and 00, we obtain the second equation 
of equilibrium 

A/> + iV>i sin <p + Zr^o = (j) 

From the two Eqs. (/) and (j) the forces 
No and N<, can be calculated in each 
particular case if the radii r and r 1 and 
the components Y and Z of the intensity 
of the external load are given. 

Instead of the equilibrium of an ele- 
ment, the equilibrium of the portion of 
the shell above the parallel circle defined by the angle ip may be considered 
(Fig. 214). If the resultant of the total load on that portion of the shell 
is denoted by R, the equation of equilibrium Is 




Fig. 214 



2-irroNip sin ip + R = 



(255) 



This equation can be used instead of the differential equation (/), from 
which it can be obtained by integration. If Eq. (J) is divided by r x r Qj 
it can be written in the form 



N t 



ri r 2 



(25C) 



It is seen that when N„ is obtained from Eq. (255), the force N e can be 
calculated from Eq. (256). Hence the problem of membrane stresses 
can be readily solved in each particular case. Some applications of these 
equations will be discussed in the next article. 



43G 



THEORY OF PLATES AND SHELLS 



106. Particular Cases of Shells in the Form of Surfaces of Revolution. 1 

Spherical Dome. Assume that a spherical shell (Fig. 215a) is submitted 
to the action of its own weight, the magnitude of which per unit area is 
constant and equal to q. Denoting the radius of the sphere by a, we 
have To = a sin <p and 

R = 2tt J a 2 q sin <p dip = 2ira 2 q(l — cos <p) 
Equations (255) and (256) then give 



N ¥ = - 



aq{\ — cos (p) 
sin 2 cp 



aq 



1 + COS (p 



N e = aql z — : cos <p ) 

* \1 + COS <p ) 

It is seen that the forces N ^ are always negative 



(257) 



There is thus a com- 
pression along the meridians that 
increases as the angle <p increases. For 
<p = we have N? = — ag/2, and 
for (p = w/2, N,p = — aq. The forces 
No are also negative for small angles <p. 
When 

1 n 

-z — : cos (p = 

1 + COS <p 

i.e. , for <p = 51°50 / , No becomes equal 
to zero and, with further increase of <p, 
becomes positive. This indicates that 
for <p greater than 51°50 / there are tensile 
stresses in the direction perpendicular 
to the meridians. 

The stresses as calculated from (257) 
will represent the actual stresses in the 
shell with great accuracy 2 if the sup- 
ports are of such a type that the reac- 
tions are tangent to meridians (Fig. 
215a). Usually the arrangement is 
such that only vertical reactions are 
imposed on the dome by the supports, 
whereas the horizontal components of the forces N^ are taken by a 

Examples of this kind can be found in the book by A. Pfluger, "Elementare 
Schalenstatik," Berlin, 1957; see also P. Forchheimer, "Die Berechnung ebener und 
gekrummter Behalterboden/ , 3d ed., Berlin, 1931, and J. W. Geckeler's article in 
"Handbuch der Physi^' vol. 6, Berlin, 1928. 

2 Small bending stresses due to strain of the middle surface will be discussed %) 
Chap. 16. 




DEFORMATION OF SHELLS WITHOUT BENDING 437 

supporting ring (Fig. 2156) which undergoes a uniform circumferential 
extension. Since this extension is usually different from the strain along 
the parallel circle of the shell, as calculated from expressions (257), 
some bending of the shell will occur near the supporting ring. An investi- 
gation of this bending 1 shows that in the case of a thin shell it is of a very 
localized character and that at a certain distance from the supporting ring 
Eqs. (257) continue to represent the stress conditions in the shell with 
satisfactory accuracy. 



/ 1 

1 ?° V 


,a 


\ 2 
1 3 


""7 


I 


f 3 sin 2 j 




~~A A 


/Vnr-$»o 


1 ^^w ' 


216 


A 

(b) 


(<0 




Fig 





Very often the upper portion of a spherical dome is removed, as shown 
in Fig. 215c, and an upper reinforcing ring is used to support the upper 
structure. If 2<p is the angle corresponding to the opening and P is the 
vertical load per unit length of the upper reinforcing ring, the resultant R 
corresponding to an angle <p is 

R = 2w I * a 2 q sin <p dip + 2irPa sin <po 

J <po 

From Eqs. (255) and (256) we then find 



AT — __ C0S ^° """ C0S ^ _ P Sin ^° 
IV <p — CLQ : x l — i -z 

sin 2 (p sin 2 <p 

%T /cos <po — cos (p \ , „ 

N = aq I *L_j — cos <p J + P 

\ sm 2 <p ) 



sin <^>o 
sin 2 <p 



(258) 



As another example of a spherical shell let us consider a spherical tank 
supported along a parallel circle A A (Fig. 216) and filled with liquid of a 
specific weight y. The inner pressure for any angle <p is given by the 

1 See Art. 131. It" should be noted, however, that in the case of a negative or zero 
curvature of the shell (rir 2 < 0) bending stresses due to the edge effect are not neces- 
sarily restricted to the edge zone of the shell. See, for instance, W. Flugge, "Statik 
und Dynamik der Schalen," p. 65, 2d ed., Berlin, 1957. The limitations of the mem- 
brane theory of shells are discussed in detail by A. L. Goldenveiser, "Theory of 
Elastic Thin Shells," p. 423, Moscow, 1953. The compatibility of a membrane state 
of stress under a given load with given boundary conditions was also discussed by 
E. Behlendorff, Z. angew. Math. Mech., vol. 36, p. 399, 1956. 



438 THEORY OF PLATES AND SHELLS 

expression 1 

p = — Z = ya(l — cos <p) 

The resultant R of this pressure for the portion of the shell defined by 
an angle <p is 

R = — 2ira 2 J ya(l — cos <p) sin <p cos <p dip 

= -27ra 3 T [i - i cos 2 <p(l - | cos.?)] 

Substituting in Eq. (255), we obtain 

*. W/- «- '<« - 2 cos '" - x (' - rfe) (259 > 

and from Eq. (256) we find that 



*r T« 2 /- « , 2 cos 2 g \ 

6 \ 1 + cos <p/ 



(260) 



Equations (259) and (260) hold for <p < <p . In calculating the resultant 
ii! for larger values of <p, that is, for the lower portion of the tank, we must 
take into account not only the internal pressure but also the sum of the 
vertical reactions along the ring A A. This sum is evidently equal to 
the total weight of the liquid 47ra 3 7/3. Hence 

R = - 4 7ra 3 T _ 2-ira s y[jr - i cos 2 <p(l - f cos <p)] 

Substituting in Eq. (255), we obtain 

Np « If ( 5 + ^os^X (261) 

* 6 \ 1 — cos <p/ 

and from Eq. (256), 

Ne „ ^.Vi _ 6 cos „ - ^f-) (262) 

6 \ 1 — cos ip) 

Comparing expressions (259) and (261), we see that along the supporting 
ring A A the forces N* change abruptly by an amount equal to 27a 2 / 
(3 sin 2 (po). The same quantity is also obtained if we consider the vertical 
reaction per unit length of the ring A A and resolve it into two compo- 
nents (Fig. 2166) : one in the direction of the tangent to the meridian and 
the other in the horizontal direction. The first of these components is 
equal to the abrupt change in the magnitude of N? mentioned above; 
the horizontal component represents the reaction on the supporting ring 
which produces in it a uniform compression. This compression can be 
eliminated if we use members in the direction of tangents to the meridians 
instead of vertical supporting members, as shown in Fig. 216a. As may 

1 A uniform pressure producing a uniform tension in the spherical shell can be 
superposed without any complication on this pressure. 



DEFORMATION OF SHELLS WITHOUT BENDING 



439 



be seen from expressions (260) and (262), the forces N e also experience 
an abrupt change at the circle A A. This indicates that there is an abrupt 
change in the circumferential expansion on the two sides of the parallel 
circle A A. Thus the membrane theory does not satisfy the condition of 
continuity at the circle A A, and we may expect some local bending to 
take place near the supporting ring. 

Conical Shell. In this case certain membrane stresses can be produced 
by a force applied at the top of the cone. If a force P is 
applied in the direction of the axis of the cone, the stress 
distribution is symmetrical, and from Fig. 217 we obtain 



N* 



(a) 




Fig. 217 



27rr cos a 

Equation (256) then gives N e = 0. The case of a force 
applied at the top in the direction of a generatrix will be 
discussed in Art. 110 and the loading of the shell by its 
weight in Art. 133. 

If lateral forces are symmetrically distributed over the conical surface, 
the membrane stresses can be calculated by using Eqs. (255) and (256). 
Since the curvature of the meridian in the case of a cone is zero, r 2 = <*> ; 

we can write these equations in the following 

form : 

R 




tf, = - 



27rTo sin <p 



N e = -Zr t = - 



Zu_ 
sin cp 



(b) 



Each of the resultant forces N<p and N$ can be 
calculated independently provided the load dis- 
tribution is known. As an example, we take the 
case of the conical tank filled with a liquid of 
specific weight y as shown in Fig. 218. Measuring the distances y from 
the bottom of the tank and denoting by d the total depth of the liquid in 
the tank, the pressure at any parallel circle ran is 

p = -Z = y(d - y) 

Also, for such a tank <p = (tt/2) + a and r = y tan a. Substituting in 
the second of the equations (6), we obtain 

y(d — y)y tan a 



Ne = 



COS a 



(c) 



This force is evidently a maximum when y = d/2, and we find 

yd 2 tan a 



(Ne\ 



4 cos a 



440 THEORY OF PLATES AND SHELLS 

In calculating the force N<, we observe that the load R in the first of the 
equations (6) is numerically equal to the weight of the liquid in the conical 
part mno together with the weight of the liquid in the cylindrical part 
must. Hence 

R = -Tryyttf - y + ly) tan 2 a 
and we obtain 

= yy(d - ft) tan a 
v 2 cos a v ' 

This force becomes a maximum when y = fd, at which point 

3 d 2 y tan a 



\±y ip)ma.x 



16 cos a 




1 



If the forces supporting the tank are in the direction of generatrices, as 
shown in Fig. 218, expressions (c) and (d) represent the stress conditions 

in the shell with great accuracy. Usually 
there will be a reinforcing ring along the 
upper edge of the tank. This ring takes the 
horizontal components of the forces N^; 
A \ dcp^f !~~ — — /A * the vertical components of the same forces 
K% n^j x I ^'' constitute the reactions supporting the 

"\ I tank. In such a case it will be found that 

a local bending of the shell takes place at 
the reinforcing ring. 

Shell in the Form of an Ellipsoid of Revolu- 
_ tion. Such a shell is used very often for the 

ends of a cylindrical boiler. In such a case 
a half of the ellipsoid is used, as shown in Fig. 219. The principal radii 
of curvature in the case of an ellipse with semiaxes a and 6 are given by 
the formulas 

= aW a? M 

Tl (a 2 sin 2 <p + & 2 cos 2 <p)* T * (a 2 sin 2 <p + 6 2 cos 2 *>)* W 

or, by using the orthogonal coordinates x and y shown in the figure, 

3 6 2 (aV + 6 4 x 2 )* ,- 

If the principal curvatures are determined from Eqs. (e) or (/), the forces 
N p and N$ are readily found from Eqs. (255) and (256). Let p be the 
uniform steam pressure in the boiler. Then for a parallel circle of a 
radius r we have R = — vpr% } and Eq. (255) gives 

N 9 = ^HL_ „ 2£ (263) 

* 2 sin <p 2 



DEFORMATION OF SHELLS WITHOUT BENDING 

Substituting in Eq. (256), we find 



N e 



= r 2 p--N^ = p[r 2 -^J 



441 



(264) 



At the top of the shell (point 0) we have r x = r 2 = cfi/b, and Eqs. (263) 
and (264) give 



At the equator A A we have r x = b 2 /a and r 2 = a; hence 



(g) 



(h) 



It is seen that the forces N v are always positive, whereas the forces Ne 
become negative at the equator if 



a 2 > 26 2 



(i) 



In the particular case of a sphere, a = b; and we find in all points 
tf, = N e = pa/2. 

>S/ieZ£ in .Porm 0/ a Torus. If a torus is obtained by rotation of a circle 
of radius a about a vertical axis (Fig. 220), the forces N? are obtained by 




Fig. 220 

considering the equilibrium of the ring-shaped portion of the shell repre- 
sented in the figure by the heavy line AB. Since the forces N 9 along the 
parallel circle BB are horizontal, we need consider only the forces N 9 
along the circle A A and the external forces acting on the ring when dis- 
cussing equilibrium in the vertical direction. Assuming that the shell is 
submitted to the action of uniform internal pressure p, we obtain the 
equation of equilibrium 



from which 



IS 



2Trr N ip sin <p = 7rp(rg — 6 2 ) 
_ V( r l — 6 2 ) _ pa(ro + b) 



2r sin <p 



2r Q 



(265) 



442 THEORY OF PLATES AND SHELLS 

Substituting this expression in Eq. (256), we find 1 

N, = E%^»> = f (266) 

A torus of an elliptical cross section may be treated in a similar manner. 

107. Shells of Constant Strength. As a first example of a shell of constant strength, 
let us consider a dome of nonuniform thickness supporting its own weight. The 
weight of the shell per unit area of the middle surface is yh, and the two components 
of this weight along the coordinate axes are 

Y = yh sin <p Z = yh cos <p (a) 

In the case of a shell of constant strength the form of the meridians is determined in 
such a way that the compressive stress is constant and equal to <r in all the directions 
in the middle surface, i.e., so that 

If, - N - -ah 
Substituting in Eq. (256), we find 



\ri r 2 / 



<*h ( 1 J = 7^ COS tp (6) 



or, by substituting r 2 = r sin <p and solving for n, 



ri «, (c) 

7 

- ro cos tp — sin ^> 



From Fig. 213b, we have 

dr 



r\ dtp = 



COS ip 

Thus Eq. (c) can be represented in the form 

dro r cos <p 



dtp y 

- ro cos ip — sin ip 
c 



(d) 



At the top of the dome where ip = 0, the right-hand side of the equation becomes 
indefinite. To remove this difficulty we use Eq. (6). Because of the conditions of 
symmetry at the top, ri = r 2 , and we conclude that 

2<r 2<r 

n =s T2 = — and dro = ri d^> = — d^> 
7 7 

1 Nevertheless, a consideration of the deformation of the shell shows that bending 
stresses inevitably must arise near the crown ro = b of the shell, and this in spite of 
the lack of any singularity either in the shape of the shell surface or in the distribu- 
tion of the loading. See W. R. Dean, Phil. Mag., ser. 7, vol. 28, p. 452, 1939, and 
also Fliigge, op. cit., p. 81. 



DEFORMATION OF SHELLS WITHOUT BENDING 



443 



Hence, for the top of the dome we have 

dr 2<r 

dif y 



(e) 



Using Eqs. (e) and (d), we can obtain the shape of the meridian by numerical inte- 
gration, starting from the top of the dome and calculating for each increment A<p 
of the angle <p the corresponding increment Ar of the radius r . To find the vari- 
ation of the thickness of the shell, Eq. (/), Art. 105, must be used. Substituting 
N<p = No = — <rh in this equation and observing that a is constant, we obtain 



d y 

(hr ) + hri cos <p -\ — riroh sin <p = 

d<p o* 

Substituting expression (c) for r h the following equation is obtained: 



(/) 



cos ip + - r sin ip 



— (hr ) = hr 
dip y 



(g) 



r cos tp — sin tp 




For <p = 0, we obtain from Eq. (/) 

— (hr ) « hri = h — 

It is seen that for the first increment A<? of the angle <p any constant value for h can 

be taken. Then for the other points of 

the meridian the thickness is found by 

the numerical integration of Eq. (g). In 

Fig. 221 the result of such a calculation 

is represented. 1 It is seen that the 

condition 

No = N* = -ah 

, n . r r Fig. 221 

brings us not only to a definite form of 

the middle surface of the dome but also to a definite law of variation of the thick- 
ness of the dome along the meridian. 

In the case of a tank of equal strength that contains a liquid with a pressure yd at 
the upper point A (Fig. 222) we must find a shape of the meridian such that an internal 
pressure equal to yz will give rise at all points of the shell to forces 2 

Ntp = No = const 

A similar problem is encountered in finding the shape of a drop of liquid resting on 
a horizontal plane. Because of the capillary forces a thin surface film of uniform 
tension is formed which envelops the liquid and prevents it from spreading over the 
supporting surface. Both problems are mathematically identical. 

1 This example has been calculated by Fluggc, op. tit., p. 38. 

2 A mathematical discussion of this problem is given in the book by C. Runge and 
H. Konig, "Vorlesungen uber numerisches Rechnen, , ' p. 320, Berlin, 1924. 



444 THEOBY OF PLATES AND SHELLS 

In such cases, Eq. (256) gives 

„ /l 1\ 

7« 






<h) 



Taking the orthogonal coordinates as shown in the figure, we have 

x dx 



ri = 



Hence 

and Eq. (h) gives 

Observing that 



2 
r 2 



sin <p 
sin tp 1 

n 



r\ dip = ds = 

COS <p 

cos ^> d^> d sin <p 



x 



d sin 



dx 



+ 



dx 

sin ^ 72 

x N 



dx 



dz 



tan ip = -7- and 
dx 



sin ^> = 



tan ^ 



Vl + tan 2 *> 



(0 



(J) 



it is possible to eliminate sin ^ from Eq. (i) and obtain in this way a differential 
equation for z as a function of x. The equation obtained in this manner is very 




&77777777777777777T77777777: 
B 'z C 

Fig. 222 

complicated, and a simpler means of solving the problem is to introduce a new vari- 
able u = sin <p. Making this substitution in Eqs. (i) and (j), we obtain 



du u __ yz 
dx x Ny 

dz 

dx 



u 



(*) 

(0 



Vi - «■ 

These equations can be integrated numerically starting from the upper point A of 
the tank. At this point, from symmetry, n =■ rt, and we find from Eq. (h) that 

2N V 



n - 



yd 



DEFORMATION OF SHELLS WITHOUT BENDING 445 

By introducing the notation 



we write 



y 



2a 2 , x 

a 



With this radius we make the first element of the meridian curve riA<p = Ax, corre- 
sponding to the small angle A<p. At the end of this arc we have, as for a small arc of 
a circle, 



d + ri — — = a + 



-h^] 



2 art , ~ _, (n) 



Ax .Ax 

u « — = a — 

ri 2a 2 



When the values w and 2 have been found from Eqs. (n), the values of du/dx and 
dz/dx for the same point are found from Eqs. (*) and (I). With these values of the 
derivatives we can calculate the values of z and u at the end of the next interval, and 
so on. Such calculations can be continued without difficulty up to an angle *> equal, 
say, to 50°, at which the value of u becomes approximately 0.75. From this point 
on and up to ^ = 140° the increments of z are much longer than the corresponding 
increments of x, and it is advantageous to take z as the independent variable instead 
of x. For ip > 140°, x must again be taken as the independent variable, and the 
calculation is continued up to point B, where the meridian curve has the horizontal 
tangent BC. Over the circular area BC the tank has a horizontal surface of contact 
with the foundation, and the pressure y(d + d x ) is balanced by the reaction of the 
foundation. 

A tank designed in this manner 1 is a tank of constant strength only if the pressure 
at A is such as assumed in the calculations. For any other value of this pressure 
the forces Ne and N* will no longer be constant but will vary along the meridian. 
Their magnitude can then be calculated by using the general equations (255) and 
(256). It will also be found that the equilibrium of the tank requires that vertical 
shearing forces act along the parallel circle BC. This indicates that close to this circle 
a local bending of the wall of the tank must take place. 

108. Displacements in Symmetrically Loaded Shells Having the Form 
of a Surface of Revolution. In the case of symmetrical deformation of a 
shell, a small displacement of a point can be resolved into two compo- 
nents: v in the direction of the tangent to the meridian and w in the 
direction of the normal to the middle surface. Considering an element 
AB of the meridian (Fig. 223), we see that the increase of the length of 
the element due to tangential displacements v and v + (dv/d<p) dip of its 
ends is equal to (dv/d<p) dip. Because of the radial displacements w of the 
points A and B the length of the element decreases by an amount w dip. 
The change in the length of the element due to the difference in the radial 
displacements of the points A and B can be neglected as a small quantity 

1 Tanks of this kind were constructed by the Chicago Bridge and Iron Works; see 
C. L. Day, Eng. News-Record, vol. 103, p. 416, 1929. 



446 



THEORY OP PLATES AND SHELLS 



of higher order. Thus the total change in length of the element AB due 
to deformation is 

-r- d<p — w dip 
a<p 

Dividing this by the initial length r x d<p of the element, we find the strain 

of the shell in the meridional direction to be 









W -* v 



* r\ dip 



w 



(a) 



"%V 



Considering an element of a parallel circle it 
may be seen (Fig. 223) that owing to displace- 
ments v and w the radius r of the circle 
increases by the amount 



Fig. 223 



v cos <p — w sin <p 

The circumference of the parallel circle increases in the same proportion 
as its radius; hence 

1 



€0 = — (v cos ip — w sin <p) 



or, substituting r = r 2 sin <p, 



€0 = — COt V? 

r 2 r 2 



(6) 



Eliminating w from Eqs. (a) and (6), we obtain for t; the differential 
equation 



j # cot ^ = r^ — r 2 €0 



(c) 



The strain components e„ and t e can be expressed in terms of the forces 
N v and N t by applying Hooke's law. This gives 



tv = Eh ^ N * ~ vNe ^ 



(d) 



Substituting in Eq. (c), we obtain 

^ - v cot <p = -^ [^(r x + vr 2 ) - AT # (r 2 + wO] (267) 

a<p tLh 

In each particular case the forces N v and N e can be found from the load- 
ing conditions, and the displacement v will then be obtained by integration 
of the differential equation (267). Denoting the right-hand side of this 



DEFORMATION OF SHELLS WITHOUT BENDING 447 

equation by f(<p), we write 

-^ - V COt <p = f(tp) 

The general solution of this equation is 

in which C is a constant of integration to be determined from the con- 
dition at the support. 

Take, as an example, a spherical shell of constant thickness loaded by 
its own weight (Fig. 215a). In such a case r x = r 2 = a, N p and N e are 
given by expressions (257), and Eq. (267) becomes 



av , a 2 q(l + *>) / 2 \ 

-z V COt if = p,, ( cos <p — 1 , — I 

1 ^ Eh \ 1 + cos <p/ 






The general solution (e) is then 

„ = <^^[ sin , log(1+cosrf _^_] +Csin , y, 

The constant C will now be determined from the condition that f or <p = a 
the displacement v is zero (Fig. 215a). From this condition 



= a'g(1 L + v) T 1 • _ log (1 + cos a) 1 

£"/i L 1 + cos a ° J 



(J7) 



The displacement v is obtained by substitution in expression (/). The 
displacement w is readily found from Eq. (6). At the support, where 
v = 0, the displacement w can be calculated directly from Eq. (6), with- 
out using solution (/), by substituting for e e its value from the second of 
the equations (d). 

109. Shells in the Form of a Surface of Revolution under Unsym- 
metrical Loading. Considering again an element cut from a shell by two 
adjacent meridians and two parallel circles (Fig. 224), in the general case 
not only normal forces N, and N$ but also shearing forces N v e = Ne* 
will act on the sides of the element. Taking the sum of the projections 
in the y direction of all forces acting on the element, we must add to the 
forces considered in Art. 105 the force 



dNe v 

dd 



ri dd dip (a) 



representing the difference in the shearing forces acting on the lateral 
sides of the element. Hence, instead of Eq. (/), Art. 105, we obtain the 



448 
equation 



THEORY OF PLATES AND SHELLS 



£<*.*> +*£"» 



iV> x cos (p + YriVo = 



(268) 



Considering the forces in the x direction, we must include the difference 
of the shearing forces acting on the top and bottom of the element as 




V^F 







N«+ — E 



^dp 



-xif 






"2 



Fig. 224 



given by the expression 



the force 



N 9 , ~d<pd6 + ^ r d<pdd = ^- (r N v9 ) dip dd (6) 

a<p dip dtp 



dN e 
80 



ri dd d(p 



due to variation of the force N$ and the force 

Nb 9 T\ cos ip d0 dip 



(c) 



(d) 



due to the small angle cos ip dd between the shearing forces N$^ acting on 
the lateral sides of the element. The component in x direction of the 
external load acting on the element is 



Xr ri dd dip 
Summing up all these forces, we obtain the equation 

Y ( r oN^ e ) + -^ n + Ne^rx cos ? + Xnn = 

The third equation of equilibrium is obtained by projecting the forces on 
the z axis. Since the projection of shearing forces on this axis vanishes, 



w 



(269) 



DEFORMATION OF SHELLS WITHOUT BENDING 449 

the third equation conforms with Eq. (256), which was derived for sym- 
metrical loading. 

The problem of determining membrane stresses under unsymmetrical 
loading reduces to the solution of Eqs. (268), (269), and (256) for given 
values of the components X, Y, and Z of the intensity of the external load. 
The application of these equations to the case of shells subjected to wind 
pressure will be discussed in the next article. 

110. Stresses Produced by Wind Pressure. 1 As a particular example 
of the application of the general equations of equilibrium derived in the 
previous article, let us consider the action of wind pressure on a shell. 
Assuming that the direction of the wind is in the meridian plane = 
and that the pressure is normal to the surface, we take 

X = Y = Z = p sin <p cos (a) 

The equations of equilibrium then become 



d t -ki \ . dN 



—_ (roN?) + -^ n - NeVi cos <p = 



£- (r N,e) + -^ r L + N 9f ri cos <p = (6) 

N^ro + N e ri sin <p = —pr ri sin <p cos 

By using the last of these equations we eliminate the force Ne and obtain 
the following two differential equations 2 of the first order for determining 

N p and No* = N^e*. 



"a7 + (ro^ + cot ^^ + r V = - priCOS ^ cose 

•s + (lp + L 1 cot *) ^ - J-^ = -pn sin 



Let us consider the particular problem of a spherical shell, in which 
case ri = r 2 = a. We take the solution of Eqs. (c) in the form 

Ny = jS, cos 6 Ne? = $**> sin (d) 

J The first investigation of this kind was made by H. Reissner, " Mtiller-Breslau- 
Festschrift," p. 181, Leipzig, 1912; see also F. Dischinger in F. von Emperger's 
"Handbuch fur Eisenbetonbau, ,, 4th ed., vol. 6, Berlin, 1928; E. Wiedemann, Schweiz. 
Bauztg., vol. 108, p. 249, 1936; and K. Girkmann, Stahlbau, vol. 6, 1933. Further 
development of the theory of unsymmetrical deformation is due to C. Truesdell, 
Trans. Am. MalK. Soc, vol. 58, p. 96, 1945, and Bull. Am. Math. Soc, vol. 54, p. 994, 
1948; E. Reissner, /. Math, and Phys., vol. 26, p. 290, 1948; and W. Zerna, Ingr.- 
Arch., vol. 17, p. 223, 1949. 

2 The application of the stress function in investigating wind stresses was used by 
A. Pucher, Pubis. Intern. Assoc. Bridge Structural Engrs. } vol. 5, p. 275, 1938; see also 
Art. 113, 



450 



THEORY OF PLATES AND SHELLS 



in which 8 P and S 0(p are functions of ip only. Substituting in Eqs. (c), 
we obtain the following ordinary differential equations for the determi- 
nation of these functions: 



^ + 2 cot <p S* + — ^ 
dSdti 



Sill ip 

1 



Se<p = —pa cos (p 



« 



+ 2 cot <p S e <> + - — S, = -pa 



d(p ... gm ^ 

By adding and subtracting these equations and introducing the notation 

Ui = $0, + So? U2 = &^ — Se<p (/) 

the following two ordinary differential equations, each containing only 
one unknown, are obtained: 

dUx 

-tjz cot, ip -t- j c/ 1 = — ptt^i -r uus <p; 

(0) 






- 1 + (2 cot ? + -J—) Ui = -pa(l + cos *>) 
y sin <p/ 

+ (2 cot ^ : ) U 2 = pa(l ~ COS ip) 

\ sin ip) 



Applying the general rule for integrating differential equations of the 
first order, we obtain 



f/i = 



*7 2 = 



1 + COS ip 

sin 3 ip 
1 — cos ip 



C\ + V a ( cos ^ ~~ o cos3 
C2 — pa ( cos <p — ^ cos 3 <p J 



•a 



w 



sm° <p 

where Ci and C2 are constants of integration. Substituting in Eqs. (/) 
and using Eqs. (d), we finally obtain 

cos 



N* = 



N 9p = 



sin* <p 
sin 



sm 3 <p 



^ "I o C0S * pa V * ~~ 

C\ — C2 , C\ + C2 ^^ 
i ^ cos 



,-Ieosv)] ^ 

v? + pa ( cos <p — 5 cos 3 <p J 



2 ' 2 

To determine the constants of integration Ci and C2 let us consider a 
shell in the form of a hemisphere and put ip = tt/2 in expressions (t). 
Then the forces along the equator of the shell are 



TiT Cl + C2 /, 

N? = ^ cos ^ 



N $ <p = o sm 



(J) 



Since the pressure at each point of the sphere is in a radial direction, 
the moment of the wind forces with respect to the diameter of the sphere 
perpendicular to the plane 6 = is zero. Using this fact and applying 
the first of the equations (j), we obtain 

/ A/> 2 cos dO = a 2 Ul ^° 2 / 



cos 2 Odd = Q 



DEFORMATION OF SHELLS WITHOUT BENDING 



451 



which gives 



C\ = — (?2 



(k) 



The second necessary equation is obtained by taking the sum of the com- 
ponents of all forces acting on the half sphere in the direction of the 
horizontal diameter in the plane = 0. This gives 



/2tt r tt/2 f 2tt n _ . . - , 

A^a sin d0 = — / / p sin ^ cos a 2 sin <^> sin v cos a<£> 



d0 



or 



C\ — C2 9 2 

a?r ^ = -7>a- £ ir 



(0 



From (A*) and (/) we obtain 

d = -fap C 2 = |ap 

Substituting these values for the constants in expressions {%) and using 
the third of the equations (6), we obtain 



N 6 



pa cos cos (f /0 . ox 

^7- ^ (2 — 3 cos <p + cos 3 if) 

3 sin 3 </? 



pa cos 
3 sin 3 <p 



(2 cos ^ — 3 sin 2 <p — 2 cos 4 v?) 



(m) 



tf 



0^ 



pa sin /0 o , * \ 

^r -r— ^ — (2 — 3 cos (p + cos 3 <£>) 
3 sin 3 <£> 



AA 



By using these expressions the wind stresses at any point of the shell 
can be readily calculated. If the shell is in the form of a hemisphere, 
there will be no normal forces acting 
along the edge of the shell, since 
(N^^^/2 = 0. The shearing forces 
N e <p along the edge are different from 
zero and are equal and opposite to the ^' 
horizontal resultant of the wind pres- 
sure. The maximum numerical value 
of these forces is found at the ends of the 
diameter perpendicular to the plane 
= 0, at which point they are equal 
to ±2pa/3. 

As a second application of Eqs. (c) let us consider the case of a shell 
having the shape of a circular cone and supported by a column at the 
vertex (Fig. 225). In this case the radius r x is infinitely large. For an 
element dy of a meridian we can write dy = ri dip. Hence 

d d 




Fig. 225 



In addition we have 

To = y sin a 



dip 1 dy 



drp 
dy 



= sin a V2 = y tan a 



452 THEORY OF PLATES AND SHELLS 

Substituting in Eqs. (c), we obtain for a conical shell submitted to a 
wind pressure Z = p sin <p cos the equations 



dAL . N ¥ . 1 dNe* 

-r- 2 H -\ : ~ = — p sin a cos 

by y y sin a dd ^ 

-5- - 4 = -p sin 

dy y 

The second equation can be readily integrated to obtain 



(n) 



**~ ~K^~ +c ) sin * 



(o) 



The edge of the shell y = Z is free from forces; hence the constant of 
integration in expression (o) is 



and we finally obtain 



c - vl s 

C ~ ~"3~ 



iV fly = | ** y8 sin (p) 



Substituting in the first of the equations (n), we find 



v . N v (p I 3 — y z . \ . 

- -\ = — l £ , . — h P sin a ) cos 

y \3 y 3 sin a / 



dy 
The integration of this equation gives 



, r p cos (I 3 - y 3 I 2 - y 2 2 \ 

N v = Z-. ( Q / jr— — COS 2 a J 

* sin a \ Sy 2 2y ) 



(«) 



which vanishes at the edge y = Z, as it should. The forces iNT* are 
obtained from the third of the equations (6), which gives 

N$ = — py s i n a cos ( r ) 

The expressions (p), (g), and (r) give the complete solution for the 
stresses due to wind pressure on the conical shell represented in Fig. 225. 
At the top {y = 0) the forces N 9 and N$ 9 become infinitely large. To 
remove this difficulty we must assume a parallel circle corresponding to 
a certain finite value of y along which the conical shell is fastened to the 
column. The forces N„ Ne v distributed along this circle balance the 
wind pressure acting on the cone. It can be seen that, if the radius of 
the circle is not sufficient, these forces may become very large. 

In the case of a transverse load Q applied at the top of the cone (Fig. 220a) we can 
satisfy Eqs. (n), in which the right-hand side becomes zero, by putting 

Q cos e , . 

N 9 - — — - - N 9 - (a) 

wy sin 2 « 



DEFORMATION OF SHELLS WITHOUT BENDING 



453 



it is readily verified by integration that the shearing force which results from the 
stresses N 9 for any section normal to the axis of the cone is equal to Q and that the 
moment of those stresses with respect to the axis 6 = ir/2 of this section equalizes the 
moment Qy cos a of the load. As for the stress components N$, they vanish through- 
out the shell, as ensues from the third of 
the equations (6), where we have to assume 
n = oo and p = 0. 

Should a load S act in the direction of 
the generatrix of the cone (Fig. 2266), we 
must combine the effect of both its com- 
ponents P = S cos a (Fig. 217) and Q = S 
sin a upon the forces N<p. 

The result is 

S 

N,, = (2 cos 6 - 1) 

2irro 

which yields the extreme values of S/2irro 
at 6 = and -3£/27rr at 6 = u-, 
respectively. 

111. Spherical Shell Supported at Iso- 
lated Points. 1 We begin with the general 
case of a shell having the form of a surface jp ig 226 

of revolution and consider the case when 

the forces are acting only along the edge of the shell so that X = Y = Z = 0. The 
general equations (6) of the preceding article then become 







Neri cos <p = 



a dN$ 

— (roNe^) + — - n + Ne^ri cos <p = 
o<p ad 

Nipro + N$ri sin tp = 

Let us take the solution of these equations in the form 

Ny = S,pn cos nd 

N$ = S$n cos 7i0 

iV^ = Se,pn sin n0 



(a) 



(*) 



where S^n, Sen, and S^ n are functions of <p only and n is an integer. Substituting 
expressions (6) in Eqs. (a), we obtain 



— (roS^n) + nriSo<pn — riS^ M cos ^> = 

d 

— (roSdipn) — nnSen + riSe^n cos <p = 
dip 

S<pn H — Sen = 
r 2 



(c) 



1 Fliigge, op. c#. For the application of the stress function in solving such prob- 
lems, see the paper by Pucher, op. cit. 



454 THEORY OF PLATES AND SHELLS 

Using the third of these equations, we can eliminate the function S$ n and thus obtain 



dSvn I 1 dro \ 

~f~ + ( - -7- + COt <p ) S,n + 

dip \ro dip J 

( 1 dro T\ \ 

I — 3 h — cot ip I S^„ 

\r d*> r 2 / 



^1 S^n 
n : = 

Ti Sin ip 

-j 2 - + I - 7- + - cot *> J Se*n + -r- 5 - - 
a^> \r a^? r 2 / sin <p 



In the particular case of a spherical shell 7*1 = r 2 = a, ro = a sin tp; and Eqs. (d) 
reduce to the following simple form: ^ 



dS t 



tpn 



n 



+ 2 cot ip Syn + Se* n = 



dip sin ^> 

doe<pn , ^ „ n 

— ~ + 2 COt ^ S^n + Syn = 

a^> sin ip 

Proceeding as in the preceding article, by taking the sum and the difference of Eqs. 
(e) and introducing the notation 

we obtain 



— - + ( 2 cot ip + - ) U ln - 

dtp y sin ip) 

— — + ( 2 cot ip - J (/.,„ = 

dip y sin ^> / 



(a) 



The solution of these equations is 



(cotf) („»?)' 



f/jn = Cm — Uon = Ciu — (h) 



_ -' -m- '-'Art. . 

sin z tp sin 2 ^> 



From Eqs. (/) we then obtain 

U ln + U 2 n 1 



&ipn ~"~ 



;[ c "( oot f)" + c -( taa l)"] 

t/ ln - 1/2„ 1 [\ / A» „ / A*! 
5 *- = — 2 2~^ [ Cln \ coi y ~ Ci ° V an 2) J 



(0 



If we have a shell without an opening at the top, expressions (t) must be finite for 
ip = 0. This requires that the constant of integration C Xu — 0A Substituting this in 
Eqs. (i) and using Eqs. (6), we find 



2 sin* *> \ 2/ 

■«» (. A" . 

— - — I tan - J si 
n 2 v\ 2/ 



cos n0 

0') 



No<p = — - r~ I tan ^ ) sin n0 
2 sin 

Substituting for ip the angle ^> corresponding to the edge of the spherical shell, we 
shall obtain the normal and the shearing forces which must be distributed along the 
edge of the shell to produce in this shell the forces (j). Taking, as an example, the 
case when y? = tt/2, that is, when the shell is a hemisphere, we obtain, from 



DEFORMATION OF SHELLS WITHOUT BENDING 



455 



expressions (j), 



(Nf)^*/* — 



C iH 



cos nd 



Con 



(*) 



(NeJv-w/i = y sin n0 



Knowing the stresses produced in a spherical shell by normal and shearing forces 
applied to the edge and proportional to cos nd and sin nd, respectively, we can treat 
the problem of any distribution of normal forces along the edge by representing this 



t.ttttlttt 



aNp 



7T 



Orr 



TTTTT 



(a) 



(b) 



xr 

'2 



n 



TT 



2 

HI 



'n i | j -: 



TT 






4e 



±Jt 



T0 



TT 






_± 



(c) 



Fig. 227 



distribution by a trigonometric series in which each term of the series is a solution 
similar to solution (j). 1 Take, as an example, the case of a hemispherical dome of 
radius a, carrying only its own weight of q psf and supported by four symmetrically 
located columns. If the dome is resting on a continuous foundation, the forces N v 
are uniformly distributed along the edge as shown in Fig. 227a, in which the intensity 
of force aN,p per unit angle is plotted against the angle 6. In the case of four equi- 
distant columns the distribution of reactions will be as shown in Fig. 2276, in which 
2e denotes the angle corresponding to the circumferential distance supported by each 
column. Subtracting the force distribution of Fig. 227a from the force distribution of 
Fig. 2276, we obtain the distribution of Fig. 227c, representing a system of forces in 
equilibrium. This distribution can be represented in the form of a series 



(a^) v>=3T /2 = 



i 



A n cos nd 



(I) 



n = 4,8,12, 



1 In using a series N^ = % 



1 



C 2 n cos nd for normal forces we obtain a dis- 



7i = l,2,3, ... 

tribution of these forces symmetrical with respect to the diameter 6 = 0. In the 
general case the series will contain not only cosine terms but also sine terms. The 
solutions for sine terms can be obtained in exactly the same manner as used in our 
discussion of the cosine terms. It is only necessary to exchange the places of cos nO 
and sin nd in Eqs. (6). 



456 THEORY OF PLATES AND SHELLS 

in which only the terms n = 4, 8, 12, . . . must be considered, since the diagram 
227c repeats itself after each interval of w/2 and has four complete periods in the 
angle 2ir. Applying the usual method for calculating the coefficients of series (i), 
we find 

A tl = sin (ne) 

ne 

Hence the distribution shown by diagram 227c is represented by the series 

00 

2qa 2 V^ sin ne ^ 



(aNf)^*/! = > cos n0 (m) 

e / { n 

n = 4,8,12 

Comparing each term of this series with the first of the equations (k) we conclude that 

4ga sin ne 



Cvn = — 



e n 



The stresses produced in the shell by the forces (m) are now obtained by taking a 
solution of the form (j) corresponding to each term of series (m) and then super- 
posing these solutions. In such a manner we obtain 



e sin 2 



c sin 2 



V^ sin ne / <p\ n 

n = 4,8,12,... 

Xsin ne / ^V 
n ^ an 2y 



n = 4,8,12,... , v 

(n) 



A^^ = — r - :; — ^ l-^l-^ ^ ^ an L ] Sin n 



n = 4,8,12... 



Superposing this solution on solution (257), which was previously obtained for a dome 
supported by forces uniformly distributed along the edge (Fig. 215a), we obtain 
formulas for calculating the stresses in a dome resting on four columns. It must be 
noted, however, that, whereas the above-mentioned superposition gives the necessary 
distribution of the reactive forces N p as shown in Fig. 2276, it also introduces shear- 
ing forces Ne<p which do not vanish at the edge of the dome. Thus our solution does 
not satisfy all the conditions of the problem. In fact, so long as we limit ourselves to 
membrane theory, we shall not have enough constants to satisfy all the conditions 
and to obtain the complete solution of the problem. In actual constructions a 
reinforcing ring is usually put along the edge of the shell to carry the shearing forces 
N$ p . In such ca,ses the solution obtained by the combination of solutions (257) and 
(n) will be a sufficiently accurate representation of the internal forces produced in a 
spherical dome resting on four columns. For a more satisfactory solution of this 
problem the bending theory of shells must be used. 1 

The method discussed in this article can also be used in the case of a nonspherical 
dome. In such cases it is necessary to have recourse to Eqs. (d), which can be solved 
with sufficient accuracy by using numerical integration. 2 

1 An example of such a solution is given in A. Aas-Jakobsen r s paper, lngr.-Arch., 
vol. 8, p. 275, 1937. 

2 An example of such integration is given by Fliigge, op. cit. On p. 51 of this book 
the calculation of membrane forces in an apsidal shell, due to the weight of the 



DEFORMATION OF SHELLS "WITHOUT BENDING 



457 



112. Membrane Theory of Cylindrical Shells. In discussing a cylin- 
drical shell (Fig. 228a) we assume that the generator of the shell is hori- 
zontal and parallel to the x axis. An element is cut from the shell by 
two adjacent generators and two cross sections perpendicular to the x axis, 
and its position is defined by the coordinate x and the angle <p. The 
forces acting on the sides of the element are shown in Fig. 2286. In 




(a) 







N ' + dH * x ^ 



3N X 
N x + 3X d * 



N Xf+ |^dx 



Fig. 228 



addition a load will be distributed over the surface of the element, the 
components of the intensity of this load being denoted, as before, by X, 
Y, and Z. Considering the equilibrium of the element and summing up 
the forces in the x direction, we obtain 



x r d(p dx H tt^ dip dx + Xr dip dx = 



dx 



dip 



(a) 



Similarly, the forces in the direction of the tangent to the normal cross 
section, i.e., in the y direction, give as a corresponding equation of 
equilibrium 



a X<P r dip dx + — — dip dx + Yr dip dx = 
ox dtp 



(b) 



The forces acting in the direction of the normal to the shell, i.e., in the 

shell, is also discussed. For application of the complex variable method to the stress 
analysis in spherical shells, see F. Martin, Ingr.-Arch., vol. 17, p. 167, 1949; see also 
V. Z. Vlasov, Priklad. Mat. Mekhan., vol. 11, p. 397, 1947. 



458 THEORY OF PLATES AND SHELLS 

z direction, give the equation 

N^ d<p dx + Zr dtp dx = 



(c) 



After simplification, the three equations of equilibrium can be repre- 
sented in the following form: 



dN x 1 3N X1P = _ x 



dx 



Xip 



r dip 
ldN« 



dx 



r dip 
N. = -Zr 



(270) 



In each particular case we readily find the value of N^. Substituting 
this value in the second of the equations, we then obtain N sp by inte- 




Fig. 229 

gration. Using the value of N X¥ > thus obtained we find N x by integrating 
the first equation. 

As an example of the application of Eqs. (270) let us consider a hori- 
zontal circular tube filled with liquid and supported at the ends. 1 Meas- 
uring the angle ip as shown in Fig. 2296 and denoting by po the pressure 
at the axis of the tube, the pressure at any point is po — 7<* cos <p. We 
thus obtain 

X = Y = Z = — po + ya cos ip 



(d) 



Substituting in Eqs. (270), we find 



N x 



N xv 

— / 7 COi 



Nj = pod — 7a 2 cos ip 
— /7a sin ip dx + Ci(ip) = — yax sin ip + Ci(<p) 



(e) 
U) 



y cos (pxdx 



_ 1 C dCM 
a J dip 



dx + C2(tp) 



x* 



= T-2 cos^-- 



x d Cijip) 
dip 



+ CM (g) 



The functions Ci(ip) and C 2 (<p) must now be determined from the con- 
ditions at the edges. 

Let us first assume that there are no forces N x at the ends of the tube. 

1 This problem was discussed by D. Thoma, Z. ges. Turbinenwesen, vol. 17, p. 49, 
1920. 



DEFOKMATION OF SHELLS WITHOUT BENDING 459 

Then 

(N x ) x =o = CN X ) X ^ = 

We shall satisfy these conditions by taking 

C 2 (<p) = CM = °^ sin v + C 

It is seen from expression (/) that the constant C represents forces N x<p 
uniformly distributed around the edge of the tube, as is the case when 
the tube is subjected to torsion. If there is no torque applied, we must 
take C = 0. Then the solution of Eqs. (270) in our particular case is 

Ny = pod — 7a 2 cos tp 
N x<p = ya f ^ "~~ x ) si* 1 V (271) 

JV* = — ~ x(Z — X) COS v? 

It is seen that N Xip and iV x are proportional, respectively, to the shearing 
force and to the bending moment of a uniformly loaded beam of span I 
and can be obtained by applying beam formulas to the tube carrying a 
uniformly distributed load of the magnitude 1 ira 2 y per unit length of the 
tube. 

By a proper selection of the function C 2 (<£>) we can also obtain a solu- 
tion of the problem for a cylindrical shell with built-in edges. In such a 
case the length of the generator remains unchanged, and we have the 
condition 



Substituting 



f* (N x - vNJ dx = 



N x = — - x(l — x) cos ip + C 2 (<p) N? = p Q a — 7a 2 cos <p 



we obtain 



(5 - -) 



C<t(if) = vp a + ( To — va 2 )y cos v 



yx ( I 2 \ 

and N x = ^- (/ — x) cos <p + vp^a + f — — *>a 2 1 7 cos <p (272) 

Owing to the action of the forces N^ and N x there will be a certain amount 
of strain in the circumferential direction at the end of the tube in contra- 
diction to our assumption of built-in edges. This indicates that at the 
ends of the tube there will be some local bending, which is disregarded in 
the membrane theory. A more complete solution of the problem can be 
obtained only by considering membrane stresses together with bending 
stresses, as will be discussed in the next chapter. 

1 The weight of the tube is neglected in this discussion. 



460 



THEORY OF PLATES AND SHELLS 



Sections of cylindrical shells, such as shown in Fig. 230, are sometimes 
used as coverings of various kinds of structures. These shells are usually 
supported only at the ends while the edges AB and CD are free. In 
calculating the membrane stresses for such shells Eqs. (270) can again be 
used. Take, for example, a shell of a semicircular cross section sup- 
porting its own weight, which is assumed to be uniformly distributed 

over the surface of the shell. In 
such a case we have 

X = Y = p sin <p Z = p cos <p 

The third of the equations (270) gives 




Ny = —pa cos <p 



w 



which vanishes along the edges AB 
and CD, as it should. It is seen that 
this condition will also be satisfied if 
some other curve is taken instead of 
a semicircle, provided that <p = + ir/2 

at the edges. Substituting expression (h) in the second of the equations 

(270), we find 



N Xip = — 2px sin (p + Ci(<p) 



(0 



By putting the origin of the coordinates at the middle of the span and 
assuming the same end conditions at both ends, x = + 1/2 of the tube, 
it can be concluded from symmetry that Ci(<p) = 0. Hence 

N Xip = -2px sin <p (J) 

It is seen that this solution does not vanish along the edges AB and CD 
as it should for free edges. In structural applications, however, the edges 
are usually reinforced by longitudinal members strong enough to resist 
the tension produced by shearing force (j). Substituting expression (j) 
in the first of the equations (270), we obtain 



7)X 

N x = *— cos ip + C 2 (<p) 

Of 



(*) 



If the ends of the shell are supported in such a manner that the reactions 
act in the planes of the end cross sections, the forces N x must vanish at 
the ends. Hence C 2 (<p) = — (pi 2 cos <p)/4a, and we obtain 



N.-- ^^ (I* - 4x*) 



(I) 



Expressions (h), (j) 9 and (I) represent the solution of Eqs. (270) for our 
particular case (Fig. 230) satisfying the conditions at the ends and also 
Ofte of the conditions along the edges AB and CD. The second con- 



DEFORMATION OF SHELLS WITHOUT BENDING 



461 



dition, which concerns the shearing forces N xv , cannot be satisfied by- 
using the membrane stresses alone. In practical applications it is 
assumed that the forces N x<f will be taken by the longitudinal members 
that reinforce the edges. It can be expected that this assumption will be 
satisfactory in those cases in which the length of the shell is not large, 
say I ^ 2a, and that the membrane theory will give an approximate 
picture of the stress distribution in such cases. For longer shells a satis- 
factory solution can be obtained only by considering bending as well as 
membrane stresses. This problem will be discussed in the next chapter 
(see Arts. 124 and 126). 




Fig. 231 

113. The Use of a Stress Function in Calculating Membrane Forces of Shells. 

In the general case of a shell given by the equation z = f(x,y) of its middle surface 
the use of a stress function 1 defining all three stress components may be convenient. 
Let us consider an element of a shell submitted to a loading the magnitude of which 
per unit area in xy plane is given by its components X, Y, Z (Fig. 231). The static 
equilibrium of the element then can be expressed by the equations 



dN x . dN v 



dx \ dx 



+ N X 



dx 


+ - 


dy 


-t- a 


= u 


dN xv 
dx 


+ 


dN y 
dy 


+ Y 


= 


,?) 


+ 


U 


*.* 


+ N. 



dz\ 

'"dx) 



+ Z = 



(a) 



(6) 



1 The introduction of the function considered here is due to A. Pucher, op. cit., and 
Beton u. Eisen, vol. 33, p. 298, 1934; see also Proc. Fifth Intern. Congr. Appl. Mech., 
Cambridge, Mass., 1938. Cylindrical coordinates, more suitable for shells in the form 
of a surface of revolution, are also used by Pucher. For a general theory of defor- 
mation following Pucher's approach, see W. Fliigge and F. Geyling, Proc. Ninth Intern. 
Congr. Appl. Mech., vol. 6, p. 250, Brussels, 1957. 



462 THEORY OF PLATES AND SHELLS 

in which the following notation is used: 



N x = N x 



cos 6 
cos <p 



N v = #, 



COS <p 
COS 



N xy = n, 



xy 



(c) 



where tan <p = dz/dx and tan = dz/dy. Carrying out the differentiation as indi- 
cated in Eq. (6) and taking into account Eqs. (a), we obtain 



- d 2 z - d 2 z 

N x — + 2N xy 

dx 2 dx dy 



d 2 z 



dz 



+ N V —--Z + X— + Y 



dz 



(d) 



dy 2 dx dy 

We can satisfy both equations (a) by introducing a stress function F(x,y) such that 

d 2 F I _ , „-„ d 2 F 



dy 2 J 



Xdx 



N u = 



dx 2 



-I 



Ydy 



N xu = - 



dxdy 



(e) 



the lower and the upper limits of the integrals being x , x and y , y } respectively, with 
x and y fixed. Substituting this in Eq. (d) we obtain the following differential equa- 
tion governing the stress function F: 



d 2 F d 2 z d 2 F d 2 z d 2 F d 2 z _ 

'dx 2 dy 2 ~~ dx dy dx dy dy 2 dx 2 ~~ * 



in which the following abbreviation is used: 

dz „ dz d 2 z 
g= -Z+X— + Y— + — 
" dx dy dx 2 



I Xdx+—I 

J W J 



Ydy 



(/) 



(9) 



If the membrane forces on the boundary of the shell are given, the respective 
boundary conditions can readily be expressed by means of Eqs. (e). If, in particular, 
the edge is connected with a vertical wall whose flexural rigidity is negligible or if the 

edge is free, then the edge forces normal to 
the elements ds of the boundary and pro- 
portional to d 2 F/ds 2 must vanish. Hence 
the variation of the stress function along 
such an edge must follow a linear law. 

A Shell in the Form of an Elliptic Parabo- 
loid. To illustrate the application of the 
method, let us take a shell in the form of an 
elliptic paraboloid (Fig. 232) with the middle 
surface 




hi h'2 



(h) 



Fig. 232 
level curves are ellipses. 



where hi and /12 are positive constants. 
The sections x = constant and y = constant 
then yield two sets of parabolas, and the 
Assuming solely a vertical load uniformly distributed over 



the ground plan of the shell and using Eqs. (/) and (g) we obtain 

1 d 2 F 2_ <W _ P 
h'2'dx 2 hxdy 2 ~~ ~~ 2 



(0 



where p *» Z is the intensity of the load. 



DEFORMATION OF SHELLS WITHOUT BENDING 



463 



Let the shell be supported by four vertical walls x = ±a/2, y = ±6/2 in such a 
way that the reactive forces normal to the respective wall vanish along the boundary. 
Consequently, the boundary conditions for the function F are d 2 F/dy 2 =0 on 
x = ± a /2 and d 2 F/dx 2 = on y = ±6/2. Thus F may be a linear function in 
x and y on the boundary. Since terms linear in x or y have no effect on stresses [see 
Eqs. (e)], this is equivalent to the condition F = on the whole boundary. 

We satisfy Eq. (i) and make F = on y = ±6/2 by taking for F the expression 



eo 

Phi (b* \ , V a x. nirX niry 

F - 4 U ~ y ) + If An cosh ~ cos T" 



(j) 



n = 1,3,5 



in which c = 6 <y/hi/h 2 . In order to fulfill the remaining condition F = on 
x ^ + a /2, we first develop the algebraic term in expression (j) into the Fourier series 



ph 
~4 



'-($-*)-*? I ;<-)-»— x 



(*) 



n = 1,3,5,... 



Substituting this in Eq. (j), making x = ±a/2, and equating the result to zero we 
obtain for each n = 1, 3, 5, . . . the equation 



2pfei6 2 



7l7ra 



(_l)(n-l)/2 _|_ ^ n cos h — - = 



7r 3 n 3 2c 

This yields the value of the coefficient A n and leads to the final solution 

nirX 



(0 



F = 



p/ii 



-r - y* -\ — r 



I 

n = l,3,5,.., 



cosh 
1 c nwy 

(__l)(n+l)/2 C oS — - 

n 3 n7ra 6 

cosh — — 
2c 



(m) 



To obtain the membrane forces we have only to differentiate this in accordance 
with the expressions (e) and to make use of the relations (c). The result is 



N> 



j)h 2 \h\ 



+ 4x 2 



+ 4y 2 



00 

1 + ^ z^ 



n7rX 

cosh 

1 c nwy 

(_l)(n+l)/2 COS — - 

n . n-ira 6 



n = 1,3,5,... 



cosh 



2c 



"—*&%& t, 



_ nirX 

cosh 

1 c ri7ri/ 

(_l)<« + l)/» cos — - 

n . ri7ra 6 



(n) 



n = 1,3,5,... 



cosh — - 
2c 



v~"\ sinh 

w +—^ cosh 



ri7ra: 



71=1,3,5,.. 



c . nxi/ 

sin — — 

nwa o 

"2c" 



All series obtained above are convergent, the only exception being the last series, 
which diverges at the corners x = ±a/2, y = ±6/2. This fact is due to a specific 



464 



THEORY OF PLATES AND SHELLS 



property of the shell surface under consideration obtained by translation of a plane 
curve. The elements of such a surface are free of any twist, and for this reason the 
membrane forces N xy fail to contribute anything to the transmission of the normal 
loading of the shell. As the forces N x and N y both vanish at the corner points of the 
shell, the shearing forces N xy near these points have to stand alone for the transmission 
of the loading. Owing to the zero twist of the surface of the shells, this leads to an 
infinite increase of those shearing forces toward the corners of the shell. Practically, 
bending moments and transverse shearing forces will arise in the vicinity of the 
corners, should the edge conditions N x = 0, N y = be fulfilled rigorously. 




Fig. 233 

A Shell in the Form of a Hyperbolic Paraboloid. 1 Another case where Pucher's 
method may be applied to advantage is a shell with a middle surface given by the 
equation 

_ _ xy 

c 

in which c = a 2 /h (Fig. 233). Hence 



(o) 



dz 


y 


dz 


X 


d 2 z d 2 z 

dx 2 " dy 2 ~~ 


d 2 z 1 


dx 


c 


dy 


c 


dx dy c 



<p) 



Provided we have to deal with a vertical loading only, the differential equation (/) 

becomes 

2 d 2F 

= -Z («) 



which yields the result 



cdx dy 



N =^ C 

-W XV — ~ 



(r) 



Let us consider first a load of an intensity Z = q, uniformly distributed over the 
horizontal projection of a shell with edges free of normal forces. Then we have 



N xy = ^ N x - N y = 



GO 



Now consider the effect of the own weight of the shell equal to qo = constant per 

1 See F. Aimond, Ginie civil, vol. 102, p. 179, 1933, and Proc. Intern. Assoc. Bridge 
Structural Engrs., vol. 4, p. 1, 1936; also B. Laffaille, Proc. Intern. Assoc. Bridge 
Structural Engrs., vol. 3, p. 295, 1935. Various cases of loading were discussed by 
K. G. Tester, Ingr.-Arch., vol. 16, p. 39, 1947. 



DEFORMATION OF SHELLS WITHOUT BENDING 
unit area of the surface. To this area corresponds an area 



465 



COS 7 = 



Vx 2 + y 2 + c 2 
of the horizontal projection of the shell. Hence 



Z = - Vx 2 + y 2 + c 2 
c 



(0 



and Eq. (r) yields 



N x 



Qo 



Vx 2 + y 2 + c 2 



Differentiating this with respect to y and then integrating the result with respect to x, 
or vice versa, both in accordance with Eqs. (e), we get 



qoy , x + Vx 2 + y 2 + c 2 

N x = - — log -. 

2 V2/ 2 + c 2 



g x . y + Vx 2 + y 2 + c 2 

Ny = - — lOg . 

2 Va: 2 + c 2 

The true forces JV» and N y are obtained from those expressions by means of Eqs. (c), 
in which the angles <p, 6 are given by tan <p = — y/c and tan 6 = — x/c. 




Fig. 234 

Several shells of this kind may be combined to form a roof, such as shown in Fig. 
234. It should be noted, however, that neither the dead load of the groin members, 
needed by such a roof, nor a partial loading — due, for instance, to snow — can be 
transmitted by the membrane forces alone; hence flexural stresses will necessarily 
arise. 1 

Of practical interest and worthy of mention are also the conoidal shells, which some- 
times have been used in the design of cantilever roofs and dam walls. 2 Roof shells of 
this kind, however, with curved generatrices instead of straight ones, have also been 
used in structural applications. 3 

1 See Flugge, op. cit., p. 119, Flugge and Geyling, op. cit., and F. A. Gerard, Trans. 
Eng. Inst. Canada, vol. 3, p. 32, 1959. 

2 The theory of the conoidal shell has been elaborated by E. Torroja, Riv. ing., 
vol. 9, p. 29, 1941. See also M. Soare, Bauingenieur, vol. 33, p. 256, 1958, and Flugge, 
op. cit., p. 127. 

3 See I. Doganoff, Bautechnik, vol. 34, p. 232, 1957. 



CHAPTER 15 

GENERAL THEORY OF CYLINDRICAL SHELLS 



114. A Circular Cylindrical Shell Loaded Symmetrically with Respect 
to Its Axis. In practical applications we frequently encounter problems 
in which a circular cylindrical shell is submitted to the action of forces 
distributed symmetrically with respect to the axis of the cylinder. The 
stress distribution in cylindrical boilers submitted to the action of steam 
pressure, stresses in cylindrical containers having a vertical axis and sub- 
mitted to internal liquid pressure, and stresses in circular pipes under 
uniform internal pressure are examples of such problems. 







t\v*. 



N* T d% 

\ 
\ 

a 



dx 






Fig. 235 



To establish the equations required for the solution of these problems 
we consider an element, as shown in Figs. 228a and 235, and consider the 
equations of equilibrium. It can be concluded from symmetry that the 
membrane shearing forces N x<p = N^ vanish in this case and that forces 
N p are constant along the circumference. Regarding the transverse 
shearing forces, it can also be concluded from symmetry that only the 
forces Q z do not vanish. Considering the moments acting on the ele- 
ment in Fig. 235, we also conclude from symmetry that the twisting 
moments M x<p = M ^ vanish and that the bending moments M^ are con- 
stant along the circumference. Under such conditions of symmetry 

466 



GENERAL THEORY OF CYLINDRICAL SHELLS 467 

three of the six equations of equilibrium of the element are identically 
satisfied, and we have to consider only the remaining three equations, 
viz., those obtained by projecting the forces on the x and z axes and by 
taking the moment of the forces about the y axis. Assuming that the 
external forces consist only of a pressure normal to the surface, these 
three equations of equilibrium are 

dN x 



dx 
dQx a dx d^ + A^ dx dip + Za dx d<p = (a) 



—t-^ a dx d<p — Q x a dx dip = 
dx 

The first one indicates that the forces N x are constant, 1 and we take them 
equal to zero in our further discussion. If they are different from zero, 
the deformation and stress corresponding to such constant forces can be 
easily calculated and superposed on stresses and deformations produced 
by lateral load. The remaining two equations can be written in the 
following simplified form: 

§k + Itf, = -Z 
dx a ., v 

*± - Q, - 

dx 

These two equations contain three unknown quantities : N^, Q x , and M x . 
To solve the problem we must therefore consider the displacements of 
points in the middle surface of the shell. 

From symmetry we conclude that the component v of the displace- 
ment in the circumferential direction vanishes. We thus have to con- 
sider only the components u and w in the x and z directions, respectively. 
The expressions for the strain components then become 



_ du __ __ w 

ax a 



(c) 



Hence, by applying Hooke's law, we obtain 

Eh . , . Eh (du w\ 

Ar Eh , , s Eh ( w du\ 

From the first of these equations it follows that 



(d) 



du _ w 
dx a 

1 The effect of these forces on bending is neglected in this discussion. 



468 THEORY OF PLATES AND SHELLS 

and the second equation gives 

v a 

Considering the bending moments, we conclude from symmetry that 
there is no change in curvature in the circumferential direction. The 
curvature in the x direction is equal to —d 2 w/dx 2 . Using the same equa- 
tions as for plates, we then obtain 



where D = 



C = vM x 

dMv (/) 

dx 2 
Eh* 



M x = -D 



12(1 - *») 



is the flexural rigidity of the shell. 

Returning now to Eqs. (b) and eliminating Q x from these equations, 
we obtain 

d 2 M 1 
dx 2 a 

from which, by using Eqs. (e) and (/), we obtain 

d* ( n d 2 w\ ,Eh „ , 07Q . 

dx\ D dx-*) + l£ W = Z (273) 

All problems of symmetrical deformation of circular cylindrical shells 
thus reduce to the integration of Eq. (273). 

The simplest application of this equation is obtained when the thick- 
ness of the shell is constant. Under such conditions Eq. (273) becomes 

»£+!?-* <™ 

Using the notation 

04 = llL . 3(1 - ,«) (275) 

p 4a 2 D a 2 /i 2 K ' 

Eq. (274) can be represented in the simplified form 

£ + ** - 1 ™ 

This is the same equation as is obtained for a prismatical bar with a 
flexural rigidity D, supported by a continuous elastic foundation and 
submitted to the action of a load of intensity Z.* The general solution 
of this equation is 

w = eP*{C\ cos fix + <7 2 sin fix) 

+ er+*(Ct cos fix + C 4 sin 0x) + /(x) (277) 

* See S. Timoshenko, "Strength of Materials," part II, 3d ed., p 2, 1956. 



GENERAL THEORY OF CYLINDRICAL SHELLS 



469 



in which f(x) is a particular solution of Eq. (276), and C h . . . , C 4 are 
the constants of integration which must be determined in each particular 
case from the conditions at the ends of the cylinder. 

Take, as an example, a long circular pipe submitted to the action of 
bending moments M and shearing forces Q , both uniformly distributed 
along the edge x = (Fig. 236). In this case 
there is no pressure Z distributed over the sur- 
face of the shell, and f(x) = in the general solu- 
tion (277). Since the forces applied at the end 
x = produce a local bending which dies out 
rapidly as the distance x from the loaded end 
increases, we conclude that the first term on 
the right-hand] side of Eq. (277) must vanish. 1 
Hence, Ci = C 2 = 0, and we obtain 



Oo 



G 



M, 



r— 



w = e-fi'iCz cos fix + Ca sin fix) 



(</) 



Fig. 236 



The two constants C 3 and d can now be determined from the conditions 
at the loaded end, which may be written 



Substituting expression (g) for w, we obtain from these end conditions 



(h) 



C a = - 



2/3 3 Z> 



(Qo + Wo) 



C 4 = 



Mo 
2$W 



(0 



Thus the final expression for w is 



w = 



[j8M (sin (3x — cos fix) — Q cos px] 



2/3 3 Z) 



(278) 



The maximum deflection is obtained at the loaded end, where 



(w) x== o = — 



1 



2/3 3 Z) 



(^M + Qo) 



(279) 



The negative sign for this deflection results from the fact that w is taken 
positive toward the axis of the cylinder. The slope at the loaded end is 

1 Observing the fact that the system of forces applied at the end of the pipe is a 
balanced one and that the length of the pipe may be increased at will, this follows also 
from the principle of Saint-Venant; see, for example, S. Timoshenko and J. N. Goodier, 
"Theory of Elasticity," 2d ed., p. 33, 1951. 



470 THEORY OF PLATES AND SHELLS 

obtained by differentiating expression (278). This gives 

/ dw\ c — ^ x 

\dx) x o = WD ^ Mo C ° S P X + ^ C0S ^ + Sin P x ^*-° 



1 



2fi 2 D 



(2fiM + Qo) (280) 



By introducing the notation 

<p(fix) = e-0 x (cos fix + sin fix) 

rf/(fix) = e-" x (cos fix — sin /3x) 

0(#r) = er+* cos /3a; 

f (fix) = e-"* sin #c 



(281) 



the expressions for deflection and its consecutive derivatives can be 
represented in the following simplified form: 



w= - 2^d IfiM^ifix) + Qod(fix)) 

dw _ 1 

dx ~ 2fiW 

d 2 w 1_ 

dx 2 ~ 2fiD 

d^w 1 

^5 - 5 [2/3M f (fix) - QoHfix)] 



[2fiM 6(fix) + Qo<p(fix)] 
[2fiM <p(fix) + 2Q f 08x)J 



(282) 



The numerical values of the functions <p(fix), f(fix), 6 (fix), and f (fix) are 
given in Table 84. 1 The functions <p(fix) and 4>(fix) are represented graph- 
ically in Fig. 237. It is seen from these curves and from Table 84 

-0.4 
-0.2 

— -f ^— ^=5s==~£x 

0.2 

0.4 

0.6 

0.8 

1.0 

1 2 3 4 5 

Fig. 237 

that the functions defining the bending of the shell approach zero as the 
quantity fix becomes large. This indicates that the bending produced in 
the shell is of a local character, as was already mentioned at the beginning 
when the constants of integration were calculated. 

If the moment M x and the deflection w are found from expressions 







J 


-* 


















/ 


s 












ss 








f 














A 


f 




■? 
















/ 






















/ 






















V 























1 The figures in this table are taken from the book by H. Zimmermann, 
Berechnung des Eisenbahnoberbaues," Berlin, 1888. 



'Die 



GENERAL. THEORY OF CYLINDRICAL SHELLS 



471 



(282), the bending moment AT, is obtained from the first of the equa- 
tions' (/), and the value of the force N v from Eq. (e). Thus all neces- 
sary information for calculating stresses in the shell can be found. 

115. Particular Cases of Symmetrical Deformation of Circular Cylin- 
drical Shells. Bending of a Long Cylindrical Shell by a Load Uniformly 
Distributed along a Circular Section (Fig. 238). If the load is far enough 
from the ends of the cylinder, solution (278) can be used for each half of 



17 

(a) 




Fig. 238 



the shell. From considerations of symmetry we conclude that the value 
of Qo in this case is -P/2. We thus obtain for the right-hand portion 

(a) 



w 



~ 2fiW L 



/3M (sin fix - cos fix) + ^ cos & x 



where x is measured from the cross section at which the load is applied. 
To calculate the moment M which appears in expression (a) we use 
expression (280), which gives the slope at z = 0. In our case this slope 
vanishes because of symmetry. Hence, 



and we obtain 



2fiM - 2 = 



P 

M » = Tfi 



(b) 



Substituting this value in expression (a), the deflection of the shell 
becomes 



w = 



Pe-e 



8/3 



3^ (sin fix + cos fix) = g-^ 



<p(fix) 



(283) 



and by differentiation we find 

P 



dz ~ ^ P 8fi*D 



e~ Px sin fix = 



4/3 2 Z) 



res*) 



#w = W2 _± 



dz 2 

dh 

dz 3 



^ e~e* (sin fix - cos fix) = - ^ 



lK/&0 



(c) 



?S = ^ 3 WD e -' xcos ^ = w m) 



472 



THEORY OF PLATES AND SHELLS 



Table 84. Table of Functions <p, ^, 6, and f 



fix 


V 


+ 


e 


r 





1.0000 


1.0000 


1.0000 





0.1 


0.9907 


0.8100 


0.9003 


0.0903 


0.2 


0,9651 


0.6398 


0.8024 


0.1627 


0.3 


0.9267 


0.4888 


0.7077 


0.2189 


0.4 


0.8784 


0.3564 


0.6174 


0.2610 


0.5 


0.8231 


0.2415 


0.5323 


0.2908 


0.6 


0.7628 


0.1431 


0.4530 


0.3099 


0.7 


0.6997 


0.0599 


0.3798 


0.3199 


0.8 


0.6354 


-0.0093 


0.3131 


0.3223 


0.9 


0.5712 


-0.0657 


0.2527 


0.3185 


1.0 


0.5083 


-0.1108 


0.1988 


0.3096 


1.1 


0.4476 


-0.1457 


0.1510 


0.2967 


1.2 


0.3899 


-0.1716 


0.1091 


0.2807 


1.3 


0.3355 


-0.1897 


0.0729 


0.2626 


1.4 


0.2849 


-0.2011 


0.0419 


0.2430 


1.5 


0.2384 


-0.2068 


0.0158 


0.2226 


1.6 


0.1959 


-0.2077 


-0.0059 


0.2018 


1.7 


0.1576 


-0.2047 


-0.0235 


0.1812 


1.8 


0.1234 


-0.1985 


-0.0376 


0.1610 


1.9 


0.0932 


-0.1899 


-0.0484 


0.1415 


2.0 


0.0667 


-0.1794 


-0.0563 


0.1230 


2.1 


0.0439 


-0.1675 


-0.0618 


0.1057 


2.2 


0.0244 


-0.1548 


-0.0652 


0.0895 


2.3 


0.0080 


-0.1416 


-0.0668 


0.0748 


2.4 


-0.0056 


-0.1282 


-0.0669 


0.0613 


2.5 


-0.0166 


-0.1149 


-0.0658 


0.0492 


2.6 


-0.0254 


-0.1019 


-0.0636 


0.0383 


2.7 


-0.0320 


-0.0895 


-0.0608 


0.0287 


2.8 


-0.0369 


-0.0777 


-0.0573 


0.0204 


2.9 


-0.0403 


-0.0666 


-0.0534 


0.0132 


3.0 


-0.0423 


-0.0563 


-0.0493 


0.0071 


3.1 


-0.0431 


-0.0469 


-0.0450 


0.0019 


3.2 


-0.0431 


-0.0383 


-0.0407 


-0.0024 


3.3 


-0.0422 


-0.0306 


-0.0364 


-0.0058 


3.4 


-0.0408 


-0.0237 


-0.0323 


-0.0085 


3.5 


-0.0389 


-0.0177 


-0.0283 


-0.0106 


3.6 


-0.0366 


-0.0124 


-0.0245 


-0.0121 


3.7 


-00341 


-0.0079 


-0.0210 


-0.0131 


3.8 


-0.0314 


-0.0040 


-0.0177 


-0.0137 


3.9 


-0.0286 


-0.0008 


-0.0147 


-0.0140 



GENERAL THEORY OF CYLINDRICAL SHELLS 



473 



Table 84. Table of Functions <p, ^, 6, and f (Continued) 



Px 


<P 


+ 


e 


f 


4.0 


-0.0258 


0.0019 


-0.0120 


-0.0139 


4.1 


-0.0231 


0.0040 


-0.0095 


-0.0136 


4.2 


-0.0204 


0.0057 


-0.0074 


-0.0131 


4.3 


-0.0179 


0070 


-0.0054 


-0.0125 


4.4 


-0.0155 


0.0079 


-0.0038 


-0.0117 


4.5 


-0.0132 


0.0085 


-0.0023 


-0.0108 


4.6 


-0.0111 


0.0089 


-0.0011 


-0.0100 


4.7 


-0.0092 


0.0090 


0.0001 


-0.0091 


4.8 


-0.0075 


0.0089 


0.0007 


-0.0082 


4.9 


-0.0059 


0.0087 


0.0014 


-0.0073 


5.0 


-0.0046 


0.0084 


0.0019 


-0.0065 


5.1 


-0.0033 


0.0080 


0.0023 


-0.0057 


5.2 


-0.0023 


0.0075 


0.0026 


-0.0049 


5.3 


-0.0014 


0.0069 


0.0028 


-0.0042 


5.4 


-0.0006 


0.0064 


0.0029 


-0 0035 


5.5 


0.0000 


0.0058 


0.0029 


-0.0029 


5.6 


0.0005 


0.0052 


0.0029 


-0.0023 


5.7 


0.0010 


0.0046 


0.0028 


-0.0018 


5.8 


0.0013 


0.0041 


0.0027 


-0.0014 


5.9 


0.0015 


0.0036 


0.0026 


-0.0010 


6.0 


0.0017 


0.0031 


0.0024 


-0.0007 


6.1 


0.0018 


0.0026 


0.0022 


-0.0004 


6.2 


0.0019 


0.0022 


0.0020 


-0.0002 


6.3 


0.0019 


0.0018 


0.0018 


+0.0001 


6.4 


0.0018 


0.0015 


0.0017 


0.0003 


6.5 


0.0018 


0.0012 


0.0015 


0.0004 


6.6 


0.0017 


0.0009 


0.0013 


0.0005 


6.7 


0.0016 


0.0006 


0.0011 


0.0006 


6.8 


0.0015 


0.0004 


0.0010 


0.0006 


6.9 


0.0014 


0.0002 


0.0008 


0.0006 


7.0 


0.0013 


0.0001 


0.0007 


0.0006 



Observing from Eqs. (6) and (/) of the preceding article that 



M x = -D 



d 2 w 
dx* 



Q, = -D 



d 3 w 
dx* 



we finally obtain the following expressions for the bending moment and 
shearing force: 



M x 



40 



+(Px) Q x = - 2- B(fix) 



(284) 



474 



THEORY OF PLATES AND SHELLS 



The results obtained are all graphically represented in Fig. 239. It is 
seen that the maximum deflection is under the load P and that its value 

as given by Eq. (283) is 




Wm.x = 



Pa?p 
2Eh 



(285) 



W= 



8£ 3 D 



?m) 



dw r >. ,„,» 



M max = — 



(286) 



in 



Qx=-fe^X) 



Fig. 239 



8/3 3 Z) 

The maximum bending moment is 
also under the load and is deter- 
mined from Eq. (284) as 

4/3 

The maximum of the absolute value 
of the shearing force is evidently 
equal to P/2. The values of all 
these quantities at a certain dis- 
tance from the load can be readily- 
obtained by using Table 84. We 
see from this table and from Fig. 
239 that all the quantities that determine the bending of the shell are small 
for x > t/ p. This fact indicates that the bending is of a local character 
and that a shell of length I = 27r//3 loaded at the middle will have practi- 
cally the same maximum deflection and the same maximum stress as a very 
long shell. 

Having the solution of the problem for the case in which a load is con- 
centrated at a circular cross section, we can 
readily solve the problem of a load dis- 
tributed along a certain length of the cylinder 
by applying the principle of superposition. 
As an example let us consider the case of a 
load of intensity q uniformly distributed 
along a length I of a cylinder (Fig. 240). 
Assuming that the load is at a considerable 
distance from the ends of the cylinder, we can 
use solution (283) to calculate the deflections. 

The deflection at a point A produced by an elementary ring load of an 
intensity 1 q d£ at a distance £ from A is obtained from expression (283) 
by substituting q d£ for P and £ for x and is 





<- — I — -> 


i 




V//////W/A 


9. 




A k4t 

«-c -A<~b~>- 


* 




w//Am>, 





Fig. 240 



qd* 



e~«(cos 0£ + sin ££) 



8/3 3 Z> 

The deflection produced at A by the total load distributed over the 
1 q d{ is the load per unit length of circumference. 



GENERAL THEORY OF CYLINDRICAL SHELLS 



475 



length I is then 

w = f' 0^ e-«(cos M + sin £{) + 



qa 2 
2Eh 



0^ <r"(cos M + sin p& 

(2 — e~ pb cos fib — e~ fic cos /3c) 



The bending moment at a point A can be calculated by similar appli- 
cation of the method of superposition. 

Cylindrical Shell with a Uniform Internal Pressure (Fig. 241). If the 
edges of the shell are free, the internal pressure p produces only a hoop 
stress 



<?t 



__ pa 



and the radius of the cylinder increases by the amount 



8= ~E 



pa 1 
Eh 



(d) 



If the ends of the shell are built in, as shown in Fig. 241a, they cannot 
move out, and local bending occurs at the edges. If the length I of the 



M K- I 

TTTTTTTTTTT^ 



A 



i 



A 






LLLLLLLLLU j i 



fa) 



I 



I 




LLLlLLLLLii 



fb) 



Fig. 241 



shell is sufficiently large, we can use solution (278) to investigate this 
bending, the moment M and the shearing force Q being determined 
from the conditions that the deflection and the slope along the built-in 
edge x = (Fig. 241a) vanish. According to these conditions, Eqs. 
(279) and (280) of the preceding article become 



1 



2p z D 
1 



(jSJlf o + Qo) = 5 



2/3 2 Z) 



(20M O + Qo) = 



where 5 is given by Eq. (d). 
Solving for M and Qo, we obtain 



Mo = 2/5 2 D5 



20 2 



Qo = -4/3 3 DS = 



2 



(287) 



476 THEORY OF PLATES AND SHELLS 

We thus obtain a positive bending moment and a negative shearing force 
acting as shown in Fig. 241a. Substituting these values in expressions 
(282), the deflection and the bending moment at any distance from the 
end can be readily calculated using Table 84. 

If, instead of built-in edges, we have simply supported edges as shown 
in Fig. 2416, the deflection and the bending moment M x vanish along the 
edge Mo = 0, and we obtain, by using Eq. (279), 

Q Q = -2j3*Z>5 

By substituting these values in solution (278) the deflection at any dis- 
tance from the end can be calculated. 

It was assumed in the preceding discussion that the length of the shell 
is large. If this is not the case, the bending at one end cannot be con- 
sidered as independent of the conditions at the other end, and recourse 
must be had to the general solution (277), which contains four constants 
of integration. The particular solution of Eq. (276) for the case of uni- 
form load (Z = — p) is — p/4j8 4 D = — pa 2 /Eh. The general solution 
(277) can then be put in the following form by the introduction of hyper- 
bolic functions in place of the exponential functions : 



^ 2 
w 



TjCL 

= — ^rr + Ci sin fix sinh Bx + C2 sin px cosh jSz 
hiti 



+ C z cos fix sinh fix + C4 cos fix cosh fix (e) 

If the origin of coordinates is taken at the middle of the cylinder, as shown 
in Fig. 2416. expression (6) must be an even function of x. Hence 

C 2 = C z = (/) 

The constants Ci and C* must now be selected so as to satisfy the con- 
ditions at the ends. If the ends are simply supported, the deflection and 
the bending moment M z must vanish at the ends, and we obtain 

(»)_/» -0 (%£) =0 (g) 



Substituting expression (e) in these relations and remembering that 
C2 = C 3 = 0, we find 

7)d 2 
— 4=fr- + Ci sin a sinh a -f- d cos a cosh a = ,, v 

C\ cos a cosh a — d sin a sinh a = 

where, for the sake of simplicity, 



2 =a 



<•> 



GENERAL THEORY OF CYLINDRICAL SHELLS 477 

From these equations we obtain 



pa 



__ ^ sin a sinh a ^ pa? 2 sin a sinh a 

1 = Wi sin 2 a sinh 2 a + cos 2 a cosh 2 a Eh cos 2a + cosh 2a 

pa 2 cos a cosh a __ pa 2 2 cos a cosh a 

Ga=Z ~Eh sin 2 a sinh 2 a + cos 2 a cosh 2 a ~ #/l cos 2a + cosh 2a 



(J) 



Substituting the values ( j) and (/) of the constants in expression (e) and 
observing from expression (275) that 






we obtain 



w 



= _ P£_(l - 2 sin a sinh a sin sinh 
64Da 4 \ cos 2a + cosh 2a 



2 cos a cosh a , ^ \ m 

o i wT~ C0S & X COsh & X ) W 

cos 2a + cosh 2a / 

In each particular case, if the dimensions of the shell are known, the 
quantity a, which is dimensionless, can be calculated by means of 
notation (i) and Eq. (275). By substituting this value in expression (I) 
the deflection of the shell at any point can be found. 

For the middle of the shell, substituting x = in expression (Z), we 

obtain 

pi 4 ( 2 cos a cos h a \ ( v 

M-o = - ^^ ^1 - cos 2a + cosh 2aJ W 

When the shell is long, a becomes large, the second term in the paren- 
theses of expression (m) becomes small, and the deflection approaches 
the value (d) calculated for the case of free ends. This indicates that in 
the case of long shells the effect of the end supports upon the deflection 
at the middle is negligible. Taking another extreme case, viz., the case 
when a is very small, we can show by expanding the trigonometric and 
hyperbolic functions in power series that the expression in parentheses in 
Eq. (ra) approaches the value 5a 4 /6 and that the deflection (I) approaches 
that for a uniformly loaded and simply supported beam of length I and 
flexural rigidity D. 

Differentiating expression (J) twice and multiplying it by D, the bend- 
ing moment is found as 



7-n d*w pi 2 ( sin a sinn a , Q 

M x = -D— = - j"! ( — ^ — ■ ---*- o cosh px cos (3x 



d 2 w _ pi 2 / sin a sinh a 
dx 2 ~~ 4a 2 Vcos 2a + cosh 2a 



cos a cosh a . Q . , Q \ * \ 
sin $x smh px 1 (n) 



cos 2a + cosh 2a 



478 



THEORY OF PLATES AND SHELLS 



At the middle of the shell this moment is 



(Jf.)-o = - ?^ 



sin a sinh a 



(o) 



4a 2 cos 2a + cosh 2a 

It is seen that for large values of a, that is, for long shells, this moment 
becomes negligibly small and the middle portion is, for all practical pur- 
poses, under the action of merely the hoop stresses pa/h. 

The case of a cylinder with built-in edges (Fig. 241a) can be treated in 
a similar manner. Going directly to the final result, 1 we find that the 
bending moment M acting along the built-in edge is 



-*, __ p_ sinh 2a — sin 2a __ p 

~ W 2 sinh 2a + sin 2a ~ 20* 

/0 N sinh 2a — sin 2a 
X 2 (2a) = 



X2(2a) 



(288) 



where ^— , - . , . . 

sinh 2a + sin 2a 

In the case of long shells, a is large, the factor X2(2a) in expression (288) 
approaches unity, and the value of the moment approaches that given 
by the first of the expressions (287). For shorter shells the value of the 
factor X2(2a) in (288) can be taken from Table 85. 

Table 85 



2a 


Xi (2a) 


Xs(2a) 


X*(2a) 


0.2 


5.000 


0.0068 


0.100 


0.4 


2.502 


0.0268 


0.200 


0.6 


1.674 


0.0601 


0.300 


0.8 


1.267 


0.1065 


0.400 


1.0 


1.033 


0.1670 


0.500 


1.2 


0.890 


0.2370 


0.596 


1.4 


0.803 


0.3170 


0.689 


1.6 


0.755 


0.4080 


0.775 


1.8 


0.735 


0.5050 


0.855 


2.0 


0.738 


0.6000 


0.925 


2.5 


0:802 


0.8220 


1.045 


3.0 


0.893 


0.9770 


1.090 


3.5 


0.966 


1.0500 


1.085 


4.0 


1.005 


1.0580 


1.050 


4.5 


1.017 


1.0400 


1.027 


5.0 


1.017 


1.0300 


1.008 



Cylindrical Shell Bent by Forces and Moments Distributed along the 
Edges. In the preceding section this problem was discussed assuming 

1 Both cases are discussed in detail by I. G. Boobnov in his " Theory of Structure 
of Ships/' vol. 2, p. 368, St. Petersburg, 1913. Also included are numerical tables 
which simplify the calculations of moments and deflections. 



GENERAL THEORY OF CYLINDRICAL SHELLS 



479 



that the shell is long and that each end can be treated independently. 
In the case of shorter shells both ends must be considered simultaneously 
by using solution (e) with four constants of integration. Proceeding as 
in the previous cases, the following results can be obtained. For the 
case of bending by uniformly distributed shearing forces Q (Fig. 242a), 
the deflection and the slope at the ends are 



0) a 



= 0,x=Z — 



\ax y x== o,x=z 



2Q j3a 2 cosh 2a + cos 2a = 

Eh sinh 2a + sin 2a 
2Q /3 2 a 2 sinh 2a - sin 2a __ 



Xi(2a) 

~«y»K -• (2 a ) 
Eh sinh 2a + sin 2a ~ Eh A ' v ' 

In the case of bending by the moments M (Fig. 2426), we obtain 

2MoP*a 2 sinh 2a - sin 2a _ 2M$W 



2Q /3a 2 

Eh 
2Qpf3W 

Eh 



(289) 



=0,i=Z — 



+ 



1=0, x=l 



Eh sinh 2a + sin 2a 
4M /3 3 a 2 cosh 2a — cos 2a 
Eh sinh 2a + sin 2a 



= + 



Eh 
4M ff 3 a 2 
Eh 



X 2 (2a) 
Xs(2a) 



(290) 



(w) 

fdw\ 
\dx) 

In the case of long shells, the factors X i, X2, and xs in expressions (289) 
and (290) are close to unity, and the results coincide with those given by 



o 0j 

i 


, 


4 

f 






z 
< 


— 1 -> 


X 








<v 


f 


^ 


Qo 



t4 



M, 



M, 



2d 



^ 



-I — 



--> 



fa) 






(b) 



Mo 



Fig. 242 



expressions (279) and (280). To simplify the calculations for shorter 
shells, the values of functions xi, X2, and \z are given in Table 85. 

Using solutions (289) and (290), the stresses in a long pipe reinforced 
by equidistant rings (Fig. 243) and submitted to the action of uniform 
internal pressure p can be readily discussed. 

Assume first that there are no rings. Then, under the action of internal 
pressure, hoop stresses cr t = pa/h will be produced, and the radius of the 
pipe will increase by the amount 



5 = 



pa/ 
Eh 



Now, taking the rings into consideration and assuming that they are abso- 
lutely rigid, we conclude that reactive forces will be produced between 
each ring and the pipe. The magnitude of the forces per unit length of 



480 



THEORY OF PLATES AND SHELLS 



the circumference of the tube will be denoted by P. The magnitude of 
P will now be determined from the condition that the forces P produce a 
deflection of the pipe under the ring equal to the expansion 8 created by 
the internal pressure p. In calculating this deflection we observe that a 
portion of the tube between two adjacent rings may be considered as the 
shell shown in Fig. 242a and b. In this case Q = — £P, and the mag- 
nitude of the bending moment M under a ring is determined from the 

condition that dw/dx = at that point. 

Hence from Eqs. (289) and (290) we find 






1--H 



-» r *—t~~' n ■•■ 



. ^ y * . y i 



2a 



P(3W . 4Mpf3W . 

X 2 (2a) -i ^ — Xs(2a) = 



Eh 
from which 



Eh 



Fig. 243 



Mo = 



Px 2 (2a) 
4|8x3(2a) 



(p) 



If the distance I between the rings is large, 1 the quantity 



2a = /SZ = 



I 



\/ah 



^3(1 - v 2 ) 



is also large, the functions X2(2a) and X3(2a) approach unity, and the 
moment M approaches the value (286). For calculating the force P 
entering in Eq. (p) the expressions for deflections as given in Eqs. (289) 
and (290) must be used. These expressions give 



P/3a 2 



or 



Eh 
P/3 



Xi(2a) 



Pfia* xl(2a) 



[ (0 x 1 X 2 2 (2< 
Xi(2a) - o 



2Eh X3 (2a) 
a) J 



2 xs(2 
For large values of 2a this reduces to 

P/3a 2 



5 = 

SEh 
a* 



~ Eh 



= V 



(291) 



2Eh 



= 8 



which coincides with Eq. (285). When 2a is not large, the value of the 
reactive forces P is calculated from Eq. (291) by using Table 85. Solv- 
ing Eq. (291) for P and substituting its expression in expression (p), 
we find 



Mo = |j X2(2a) 



(292) 



This coincides with expression (288) previously obtained for a shell with 
built-in edges. 

To take into account the extension of rings we observe that the reactive 

1 For v = 0.3, 2a = 1.285Z/ y/ah. 



GENERAL THEORY OF CYLINDRICAL SHELLS 481 

forces P produce in the ring a tensile force Pa and that the corresponding 
increase of the inner radius of the ring is 1 

8l ~AE 

where A is the cross-sectional area of the ring. To take this extension 
into account we substitute 5 — 5i for 5 in Eq. (291) and obtain 

From this equation, P can be readily obtained by using Table 85, and 
the moment found by substituting p - (Ph/A) for p in Eq. (292). 

If the pressure p acts not only on the cylindrical shell but also on the 
ends, longitudinal forces 



N 



X 



__ pa 



are produced in the shell. The extension of the radius of the cylinder is 
then 



8 ~Eh 



(<-*■) 



and the quantity p{\ - $v) must be substituted for p in Eqs. (292) and 

(293). 

Equations (293) and (291) can also be used in the case of external 
uniform pressure provided the compressive stresses in the ring and in 
the shell are far enough from the critical stresses at which buckling may 
occur. 2 This case is of practical importance in the design of submarines 
and has been discussed by several authors. 3 

116. Pressure Vessels. The method illustrated by the examples of the 
preceding article can also be applied in the analysis of stresses in cylindri- 
cal vessels submitted to the action of internal pressure. 4 In discussing 
the "membrane theory" it was repeatedly indicated that this theory fails 
to represent the true stresses in those portions of a shell close to the 

1 It is assumed that the cross-sectional dimensions of the ring are small in com- 
parison with the radius o. 

2 Buckling of rings and cylindrical shells is discussed in S. Timoshenko, "Theory 

of Elastic Stability," 1936. 

s See paper by K. von Sanden and K. Gunther, " Werft und Reederei," vol. 1, 1920, 
pp. 163-168, 189-198, 216-221, and vol. 2, 1921, pp. 505-510. 

"See also M. Esslinger, "Statische Berechnung von Kesselboden," Berlin, 1952; 
G. Salet and J. Barthelemy, Bull. Assoc. Tech. Maritime Aeronaut., vol. 44, p. 505, 
1945; J. L. Maulbetsch and M. Hetenyi, ASCE Design Data, no. 1, 1944, and F. 
Schultz-Grunow, Ingr.-Arch., vol. 4, p. 545, 1933; N. L. Svensson, J. Appl. Mechanics, 
vol. 25, p. 89, 1958. 



482 



THEORY OF PLATES AND SHELLS 



edges, since the edge conditions usually cannot be completely satisfied 
by considering only membrane stresses. A similar condition in which 
the membrane theory is inadequate is found in cylindrical pressure vessels 
at the joints between the cylindrical portion and the ends of the vessel. 
At these joints the membrane stresses are usually accompanied by local 
bending stresses which are distributed symmetrically with respect to the 
axis of the cylinder. These local stresses can be calculated by using 
solution (278) of Art. 114. 

Let us begin with the simple case of a cylindrical vessel with hemi- 
spherical ends (Fig. 244). l At a sufficient distance from the joints ran 





Fig. 244 

and m\U\ the membrane theory is accurate enough and gives for the 
cylindrical portion of radius a 

pa 

T 



N x = 



N t = pa 



(a) 



where p denotes the internal pressure. 

For the spherical ends this theory gives a uniform tensile force 

N = H? 
2 



(6) 



The extension of the radius of the cylindrical shell under the action of 
the forces (a) is 



and the extension of the radius of the spherical ends is 



5 2 = 



per 

2Eh 



(1 -v) 



(c) 



(d) 



Comparing expressions (c) and (d), it can be concluded that if we con- 
sider only membrane stresses we obtain a discontinuity at the joints as 
represented in Fig. 2446. This indicates that at the joint there must act 



1 This case was discussed by E. Meissner, Schweiz. Bauztg., vol. 86, p. 1, 1925. 



GENERAL THEORY OF CYLINDRICAL SHELLS 483 

shearing forces Qo and bending moments M uniformly distributed along 
the circumference and of such magnitudes as to eliminate this discon- 
tinuity. The stresses produced by these forces are sometimes called 
discontinuity stresses. 

In calculating the quantities Qo and M we assume that the bending is 
of a local character so that solution (278) can be applied with sufficient 
accuracy in discussing the bending of the cylindrical portion. The 
investigation of the bending of the spherical ends represents a more 
complicated problem which will be fully discussed in Chap. 16. Here 
we obtain an approximate solution of the problem by assuming that the 
bending is of importance only in the zone of the spherical shell close to 
the joint and that this zone can be treated as a portion of a long cylindri- 
cal shell 1 of radius a. If the thickness of the spherical and the cylindrical 
portion of the vessel is the same, the forces Qo produce equal rotations 
of the edges of both portions at the joint (Fig. 244b). This indicates 
that Mo vanishes and that Q alone is sufficient to eliminate the discon- 
tinuity. The magnitude of Qo is now determined from the condition that 
the sum of the numerical values of the deflections of the edges of the two 
parts must be equal to the difference 5i - 5 2 of the radial expansions 
furnished by the membrane theory. Using Eq. (279) for the deflections, 
we obtain 

0W " 8l ~ 2 ~ 2Eh 
from which, by using notation (275), 

Qo ~ 2Eh 80 KJ 

Having obtained this value of the force Qo, the deflection and the bend- 
ing moment M x can be calculated at any point by using formulas (282), 
which give 2 

w = WD 6 ^ X) I 

* dx* sv ' 

Substituting expression (e) for Qo and expression (275) for in the 
formula for M x , we obtain 

Mx = _ «fr r(/te ) (/) 

8 V3(l - v 2 ) 

1 E. Meissner, in the above-mentioned paper, showed that the error in the mag- 
nitude of the bending stresses as calculated from such an approximate solution is small 
for thin hemispherical shells and is smaller than 1 per cent if a/h > 30. 

2 Note that the direction of Q in Fig. 244 is opposite to the direction in Fig. 236. 



484 



THEORY OF PLATES AND SHELLS 



This moment attains its numerical maximum at the distance x = tt/4j8, 
at which point the derivative of the moment is zero, as can be seen from 
the fourth of the equations (282). 

Combining the maximum bending stress produced by M x with the 
membrane stress, we find 




4 h V3(l 



1.293 



op 
2h 



(g) 



This stress which acts at the outer surface of the cylindrical shell is about 
30 per cent larger than the membrane stress acting in the axial direction. 
In calculating stresses in the circumferential direction in addition to the 
membrane stress pa/h, the hoop stress caused by the deflection w as well 
as the bending stress produced by the moment M^ = vM x must be con- 
sidered. In this way we obtain at the outer surface of the cylindrical shell 



ap 



Ew 
a 



-~v M - 



h J/ 4 



8(0x) + 



4 V3(l - v 2 ) 



tifix) 



Taking v = 0.3 and using Table 84, we find 



(fit). 



= 1.032^ 



atjSx = 1.85 



(h) 



Since the membrane stress is smaller in the ends than in the cylinder 
sides, the maximum stress in the spherical ends is 
always smaller than the calculated stress (h). Thus 
the latter stress is the determining factor in the design 
of the vessel. 

The same method of calculating discontinuity stresses 
can be applied in the case of ends having the form of an 
ellipsoid of revolution. The membrane stresses in this 
case are obtained from expressions (263) and (264) (see 
page 440). At the joint ran which represents the 
equator of the ellipsoid (Fig. 245), the stresses in the 

direction of the meridian and in the equatorial direction are, respectively, 




245 



<r* = 



__ pa 
~ 2h 



h y 2fey 

The extension of the radius of the equator is 

\ 2b* 2/ 



(0 



** = ~E ^ (Te ~ 



,) = 



_ V a 



Substituting this quantity for 5 2 in the previous calculation of the shear- 
ing force Qo, we find 

, _ pa 2 a 2 
8l ~ S > ~ Eh W 



GENERAL THEORY OF CYLINDRICAL SHELLS 

and, instead of Eq. (e), we obtain 



485 



p a" 
8/3 6* 



It is seen that the shearing force Q in the case of ellipsoidal ends is 
larger than in the case of hemispherical ends in the ratio o 2 /b 2 . The 
discontinuity stresses will evidently increase in the same proportion. 
For example, taking a/b = 2, we obtain, from expressions (g) and (h), 



™-* 2h ± h V3(l - ,«) 
(*)-«"• 1.128 22 



i) = 2 - 172 ! 



Again, (<r t )m* x is the largest stress and is consequently the determining 
factor in design. 1 

117. Cylindrical Tanks with Uniform Wall Thickness. If a tank is sub- 
mitted to the action of a liquid pressure, as shown in Fig. 246, the stresses 
in the wall can be analyzed by using Eq. (276). Substituting in this 
equation 

Z = - 7 (d - x) (a) 

where y is the weight per unit volume of the liquid, we obtain 
_ + ^ = __2 (6) 

A particular solution of this equation is 

Wl= ~ ~WD Eh (C) 

This expression represents the radial expansion z Qo y°| 

of a cylindrical shell with free edges under the / "///////A//////s><&// 
action of hoop stresses. Substituting expres- 
sion (c) in place oif(x) in expression (277), we obtain for the complete solu- 
tion of Eq. (b) 

y{d — x)a 2 



M„ 



w = e fix (Ci cos fix + C 2 sin fix) + e^ x (C z cos /So: + C 4 sin /3:r) — 



#/& 



In most practical cases the wall thickness h is small in comparison with 
both the radius a and the depth d of the tank, and we may consider the 
shell as infinitely long. The constants C\ and C2 are then equal to zero, 

1 More detail regarding stresses in boilers with ellipsoidal ends can be found in the 
book by Hohn, "tJber die Festigkeit der gewolbten Boden und der Zylinderschale," 
Zurich, 1927. Also included are the results of experimental investigations of dis- 
continuity stresses which are in a good agreement with the approximate solution. 
See also Schultz-Grunow, loc. cit. 



486 THEORY OF PLATES AND SHELLS 

and we obtain 

w = e-'*(C t cos fix + C 4 sin fix) - 7<<d ~*> a * (d) 

The constants C 3 and C 4 can now be obtained from the conditions at the 
bottom of the tank. Assuming that the lower edge of the wall is built 
into an absolutely rigid foundation, the boundary conditions are 

(»)_, = c 3 - ^ = 



(it) = \ -^-"'(cos fix + sin fix) 

M L-o MV z; Eh 



+ fiC*r»>(coB fix - sin flx) + ^1 = fi(C, - C.) + ^ = 



From these equations we obtain 



Ls ~~Eh L4 ~Eh 



H) 



Expression (d) then becomes 

w = — ™r- | d — a: — e"' 3 * 



d cos jSa; + f d — - ) sin fix \ 



from which, by using the notation of Eqs. (281), we obtain 

•--^^-a-'^-O-s)^] (e) 

From this expression the deflection at any point can be readily calculated 
by the use of Table 84. The force JV* in the circumferential direction is 
then 

Ehw 



N. = - 



From the second derivative of expression (e) we obtain the bending 
moment 

Having expressions (/) and (#), the maximum stress at any point can 
readily be calculated in each particular case. The bending moment has 
its maximum value at the bottom, where it is equal to 



(M x ) z ^ = Mo = (\-±\ / ?™ n = (h) 



yadh 
Vl2(l - v*j 



GENERAL THEORY OF CYLINDRICAL SHELLS 



487 



The same result can be obtained by using solutions (279) and (280) 
(pages 469, 470). Assuming that the lower edge of the shell is entirely 
free, we obtain from expression (c) 



(t»i)*_o = 






/ dwi\ _ ja? 
\ dx ) x =o Eh 



(i) 



To eliminate this displacement and rotation of the edge and thus satisfy 
the edge conditions at the bottom of the tank, a shearing force Qo and 
bending moment M must be applied as indicated in Fig. 246. The 
magnitude of each of these quantities is obtained by equating expressions 
(279) and (280) to expressions (i) taken with reversed signs. This gives 

1 (/3Mo + Qo) = + ^ 



2/3 3 D 
1 



ya £ 



(2/3M + Qo) = - 4t- 



2/3 2 jD ^"" " * ^ u/ Eh 

From these equations we again obtain expression (h) for Af , whereas for 
the shearing force we find 1 

yadh 



Qo- - 



Vl2(l 



**) V d) 



(i) 



Taking, as an example, a = 30 ft, d = 26 ft, h = 14 in., y = 0.03613 lb per in. 3, 
and v = 0.25, we find p = 0.01824 in." 1 and /3d = 5.691. For such a value of 0d our 
assumption that the shell is infinitely long results in an accurate value for the moment 
and the shearing force, and we obtain from expressions 

2a H 



(h) and (J) 

Mo = 13,960 in.-lb per in. 



Qo = -563.6 1b per in- 



m 



m, 



Fig. 247 



In the construction of steel tanks, metallic sheets of 
several different thicknesses are very often used as 
shown in Fig. 247. Applying the particular solution 
(c) to each portion of uniform thickness, we find that 
the differences in thickness give rise to discontinuities s^////////////////^//// 
in the displacement Wi along the joints ran and mini. 

These discontinuities, together with the displace- 
ments at the bottom ab, can be removed by apply- 
ing moments and shearing forces. Assuming that the vertical dimension of each 
portion is sufficiently large to justify the application of the formulas for an infinitely 
large shell, we calculate the discontinuity moments and shearing forces as before by 
using Eqs. (279) and (280) and applying at each joint the two conditions that the 
adjacent portions of the shell have equal deflections and a common tangent. If the 
use of formulas (279) and (280) derived for an infinitely long shell cannot be justified, 
the general solution containing four constants of integration must be applied to each 
portion of the tank. The determination of the constants under such conditions 
becomes much more complicated, since the fact that each joint cannot be treated 

1 The negative sign indicates that Q has the direction shown in Fig. 246 which is 
opposite to the direction used in Fig. 236 when deriving expressions (279) and (280). 



488 



THEORY OF PLATES AND SHELLS 



independently necessitates the solution of a system of simultaneous equations. This 
problem can be solved by approximate methods. 1 

118. Cylindrical Tanks with Nonuniform Wall Thickness. In the case of tanks of 
nonuniform wall thickness the solution of the problem requires the integration of 
Eq. (273), considering the flexural rigidity D and the thickness h as no longer constant 
but as functions of x. We have thus to deal with a linear differential equation of 
fourth order with variable coefficients. As an example, let us consider the case when 
the thickness of the wall is a linear function of the coordinate x. * Taking* the origin 
of the coordinates as shown in Fig. 248, we have for the thickness of the wall and for 
the flexural rigidity the expressions 



h = 



ax 



D = 



Ea* 



12(1 - „«) 



and Eq. (273) becomes 



dx 2 \ dx*) 



, 12(1 - v*) 
H — xw = 



2„2 



a £ a 



12(1 - y*)y{ X - So) 
Ea* 



The particular solution of this equation is 



cc 



r- 



^///////A////// / WA 



T J 



W\ = — 



yd* x 
Eo 



2* - 



Xo 



(a) 



(b) 



(c) 



This solution represents the radial expansion of a shell 
with free edges under the internal pressure y(x — x )- 
As a result of the displacement (c) a certain amount of 
bending of the generatrices of the cylinder occurs. 
The corresponding bending moment is 



M x = -Z> 



d 2 W! 

~~dtf 



ya 2 a 2 xo 
6(1 - v*) 



(d) 



This moment is independent of x and is in all practical 
cases of such small magnitude that its action can 
usually be neglected. 

To obtain the complete solution of Eq. (6) we have 



2a >* 

Fig. 248 
to add to the particular solution (c) the solution of the homogeneous equation 



dx*\ dx*) 



+ 



12(1 - „*) 



2/,2 



c?a 



xw = 



1 An approximate method of solving this problem was given by C. Runge, Z. Math. 
Physik, vol. 51, p. 254, 1904. This method was applied by K. Girkmann in a design 
of a large welded tank; see Stahlbau, vol. 4, p. 25, 1931. 

* H. Reissner, Beton u. Eisen, vol. 7, p. 150, 1908; see also W. Flugge, "Statik und 
Dynamik der Schalen," 2d ed., p. 167, Berlin, 1957. For tanks slightly deviating 
from the cylindrical form see K. Federhofer, Osterr. Bauzeitschrift, vol. 6, p. 149, 1951; 
and for tanks with thickness varying in accordance with a quadratic law, see Feder- 
hofer, Osterr. Ingr.-Arch., vol. 6, p. 43, 1952. A parameter method, akin to that 
explained in Art. 40, has been used by H. Faure, Proc. Ninth Congr. Appl. Mech. 
Brussels, vol. 6, p. 297, 1957. Many data regarding the design of water tanks are 
found in W. S. Gray, " Reinforced Concrete Reservoirs and Tanks," London, 1954, 
and in V. Lewe, "Handbuch fur Eisenbetonbau," vol. 9, Berlin, 1934. 



GENERAL THEORY OF CYLINDRICAL SHELLS 489 

vhich, upon division by x } can be also written 



xdx 2 \ dx 2 / 



, 12(1 - v 2 ) 

H — — w = (€) 

a 2 a 2 



The solution of this equation of the fourth order can be reduced to that of two equa- 
ions of the second order 1 if we observe that 

xdx 2 \ dx 2 / xdx\ dx\xdx\ dx J ]) 
For simplification we introduce the following symbols: 

Id/ dw\ /jPX 



12(1 - v 2 ) 
a 2 a 2 



Equation (e) then becomes 



L[L(tt;)] + P 4 w - (h) 

ind can be rewritten in one of the two following forms: 



L[L(w) + i P 2 w] - i P 2 [L(w) + i P 2 w] - 
L[L(ti>) - i P 2 w] + i P 2 [L(w) - t'p 2 w] - 

where i = V — 1. 
We see that Eq. (A) is satisfied by the solutions of the second-order equations 



(0 



L(to) + z P 2 w = (j) 

L(w) - zp 2 w = (k) 

Assuming that 

Wl = <Pl + i<P2 W2 = <PZ + *V>4 (0 

are the two linearly independent solutions of Eq. (j), it can be seen that 

w 3 = <pi — z>2 and tt>4 = <Pz — i<PA ( m ) 

are the solutions of Eq. (k). All four solutions (I) and (m) together then represent 
the complete system of independent solutions of Eq. (h). By using the sums and the 
differences of solutions (Z) and (ra), the general solution of Eq. (h) can be represented 
in the following form: 

W = C\<p\ + Cvp% + Czipz + C A iPA (*0 

in which Ci, . . . , C 4 are arbitrary constants. Thus the problem reduces to the 
determination of four functions <pi, . . . , <p*, which can all be obtained if the com- 
plete solution of either Eq. 0') or Eq. (k) is known. 

Taking Eq. (j) and substituting for L(w) its meaning (/), we obtain 

dhv 'dw t . n - f n 

x — + 2 — + *p 2 w - (o) 

dx 2 dx 

x This reduction was shown by G. Kirchhoff, "Berliner Monatsberichte," p. 815, 
1879; see also I. Todhunter and K. Pearson, " A History of the Theory of Elasticity," 
vol. 2, part 2, p. 92. 



490 THEORY OF PLATES AND SHELLS 

By introducing new variables 

rj = 2p \/ix f = w y/x (p) 



v 2 P.+V^ + (v 2 ~ l)f = (r) 



Eq. (o) becomes 

drj 2 drj 

We take as a solution of this equation the power series 

f i = a + an7 + a,2V 2 + • • • («) 

Substituting this series in Eq. (r) and equating the coefficients of each power of rj to 
zero, we obtain the following relation between the coefficients of series (s) : 

(rc 2 - l)a„ + a n - 2 = (0 

Applying this equation to the first two coefficients and taking a_i = a_2 = 0, we find 
that a = and that ai can be taken equal to any arbitrary constant. Calculating 
the further coefficients by means of Eq. (0, we find that series (s) is 

r = c' v -\i - -£- +— ^ 5! + . . .1 - cv,« («) 

* 2[ 2-4^2-4-6 2-(4-6)»-8 J 

where Ji(ij) is the Bessel function of the first kind and of the first order. For our 
further discussion it is advantageous to use the relation 

in which the series in brackets, denoted by Jo, is the Bessel function of the first kind 
and of zero order. Substituting the expression 2p \/ix for rj [see notation (p)] in 
the series representing Jo(v) and collecting the real and the imaginary terms, we 
obtain 

Ml) = *i(2p Vx) + i+ 2 (2p Vx) (w) 

where 

*i(2p Vx) = 1 — + 



(2-4)« (2.4 : 6-8)« _ (294) 

/o /^ _ (2p ^ x)1 a. (2p ^ x) * - (2p ^ X)l ° 4- 
M2 P Vx) + (2 . 4 . 6) i ( 2 • 4 • 6 • 8 • 10)' "*" " 

The solution (u) then gives 

J-i C'[^',(2p V*) + t*',(2 P y/x)] (a') 

where ^', and ^' 2 denote the derivatives of the functions (294) with respect to the 

argument 2p Vx. 

The second integral of Eq. (r) is of a more complicated form. Without derivation 
it can be represented in the form 

U = C"[*' 3 (2p Vx) + #' 4 (2 P Vx)] (60 



GENERAL THEORY OF CYLINDRICAL SHELLS 



491 



Table 86. Table of the #(x) Functions 









<tyi(x) 


# 2 (x) 


X 


+i(x) 


Mx) 


dx 


dx 


0.00 


+ 1.0000 


0.0000 


0.0000 


0.0000 


0.10 


+ 1.0000 


-0.0025 


-0.0001 


-0.0500 


0.20 


+ 1.0000 


-0.0100 


-0.0005 


-0.1000 


0.30 


+0.9999 


-0.0225 


-0.0017 


-0.1500 


0.40 


+0.9996 


-0.0400 


-0.0040 


-0.2000 


0.50 


+0.9990 


-0.0625 


-0.0078 


-0.2499 


0.60 


+0.9980 


-0.0900 


-0.0135 


-0.2998 


0.70 


+0.9962 


-0.1224 


-0.0214 


-0.3496 


0.80 


+0.9936 


-0.1599 


-0.0320 


-0.3991 


0.90 


+0.9898 


-0.2023 


-0.0455 


-0.4485 


1.00 


+0.9844 


-0.2496 


-0.0624 


-0.4974 


1.10 


+0.9771 


-0.3017 


-0.0831 


-0.5458 


1.20 


+0.9676 


-0.3587 


-0.1078 


-0.5935 


1.30 


+0.9554 


-0.4204 


-0.1370 


-0.6403 


1.40 


+0.9401 


-0.4867 


-0.1709 


-0.6860 


1.50 


+0.9211 


-0.5576 


-0.2100 


-0.7302 


1.60 


+0.8979 


-0.6327 


-0.2545 


-0.7727 


1.70 


+0 8700 


-0.7120 


-0.3048 


-08131 


1.80 


+0.8367 


-0.7953 


-0.3612 


-0.8509 


1.90 


+0.7975 


-0.8821 


-0.4238 


-0.8857 


2.00 


+0.7517 


-0.9723 


-0.4931 


-0.9170 


2.10 


+0 . 6987 


-1.0654 


-0.5690 


-0.9442 


2.20 


+0.6377 


-1.1610 


-0.6520 


-0.9661 


2.30 


+0 . 5680 


-1.2585 


-0.7420 


-0.9836 


2.40 


+0.4890 


-1.3575 


-0.8392 


-0.9944 


2.50 


+0 . 4000 


- 1 . 4572 


-0.9436 


-0.9983 


2.60 


+0.3001 


- 1 . 5569 


-1.0552 


-0.9943 


2.70 


+0.1887 


- 1 . 6557 


-1.1737 


-0.9815 


2.80 


+0.0651 


-1.7529 


-1.2993 


-0.9589 


2.90 


-0.0714 


- 1 . 8472 


-1.4315 


-0.9256 


3.00 


-0.2214 


-1.9376 


-1.5698 


-0.8804 


3.10 


-0.3855 


-2.0228 


-1.7141 


-0.8223 


3.20 


-0.5644 


-2.1016 


-1.8636 


-0.7499 


3.30 


-0.7584 


-2.1723 


-2.0177 


-0.6621 


3.40 


-0.9680 


-2.2334 


-2.1755 


-0.5577 


3.50 


-1.1936 


-2.2832 


-2.3361 


-0.4353 


3.60 


-1.4353 


-2.3199 


-2.4983 


-0.2936 


3.70 


- 1 . 6933 


-2.3413 


-2.6608 


-0.1315 


3.80 


-1.9674 


-2.3454 


-2.8221 


+0.0526 


3.90 


-2.2576 


-2.3300 


-2.9808 


+0.2596 



492 



THEORY OF PLATES AND SHELLS 



Table 86. Table of the \f/(x) Functions (Continued) 









d$i(x) 


tyiix) 


X 


Mx) 


4> 2 (x) 


dx 


dx 


4.00 


-2.5634 


-2.2927 


-3.1346 


+0.4912 


4.10 


-2.8843 


-2.2309 


-3.2819 


+0.7482 


4.20 


-3.2195 


-2.1422 


-3.4199 


+1.0318 


4.30 


-3.5679 


-2.0236 


-3.5465 


+1.3433 


4.40 


-3.9283 


-1.8726 


-3.6587 


+1.6833 


4.50 


-4.2991 


-1.6860 


-3.7536 


+2.0526 


4.60 


-4.6784 


-1.4610 


-3.8280 


+2.4520 


4.70 


-5.0639 


-1.1946 


-3.8782 


+2.8818 


4.80 


-5.4531 


-0.8837 


-3.9006 


+3.3422 


4.90 


-5.8429 


-0.5251 


-3.8910 


+3.8330 


5.00 


-6.2301 


-0.1160 


-3.8454 


+4.3542 


5.10 


-6.6107 


+0.3467 


-3.7589 


+4.9046 


5.20 


-6.9803 


+0.8658 


-3.6270 


+5.4835 


5.30 


-7.3344 


+1.4443 


-3.4446 


+6.0893 


5.40 


-7.6674 


+2.0845 


-3.2063 


+6.7198 


5.50 


-7.9736 


+2.7890 


-2.9070 


+7.3729 


5.60 


-8.2466 


+3.5597 


-2.5409 


+8.0453 


5.70 


-8.4794 


+4.3986 


-2.1024 


+8.7336 


5.80 


-8.6644 


+5.3068 


-1.5856 


+9.4332 


5.90 


-8.7937 


+6.2854 


-0.9844 


+10.1394 


6.00 


-8.8583 


+7.3347 


-0.2931 


+10.3462 



in which 4/' t and ^/\ are the derivatives with respect to the argument 2p s/x of the 
following functions : 



/- 1 r- 2 T B2p Vx i- 

*,(2p Vx) = - *,(2 P Vx) - - \Ri + log ^ M2 P Vx) 

/- 1 /- 2 T 02p Vx r- 

* 4 (2 P Vx) = - ^ 2 (2 P Vx) + - i? 2 + log £-£- ^,(2p Vx) 

2 if L ^ 



(295) 



where 



/ 2p Vx\ 5(3) / 2p V^ 6 8(5) hp Vx^ 10 

1 \ 2 / (3-2)«\ 2 / + (5-'4-3-2)'\ 2 / 

R = Ml ( 2 p^ \ _ s w (?vVxY 

8 2' \ 2 / (4-3-2)»\ 2 / 



+ 



5(6) 



(6 • 5 • 4 • 3 • 2)* 




log = 0.57722 



GENERAL THEORY OF CYLINDRICAL SHELLS 



493 



Table 86. Table of the \p(x) Functions {Continued) 









diPz(x) 


ety 4 (x) 


X 


iAj(x) 


*«(*) 


dx 


dx 


0.00 


+0.5000 


00 


0.0000 


+ 00 


0.10 


+0.4946 


-1.5409 


-0.0929 


+6.3413 


0.20 


+0.4826 


-1.1034 


-0.1419 


+3.1340 


0.30 


+0.4667 


-0.8513 


-0.1746 


+2.0498 


0.40 


+0.4480 


-0.6765 


-0.1970 


+1.4974 


0.50 


+0.4275 


-0.5449 


-0.2121 


+1.1585 


0.60 


+0.4058 


-0.4412 


-0.2216 


+0.9273 


0.70 


+0.3834 


-0.3574 


-0.2268 


+0.7582 


0.80 


+0.3606 


-0.2883 


-0.2286 


+0.6286 


0.90 


+0.3377 


-0.2308 


-0.2276 


+0.5258 


1.00 


+0.3151 


-0.1825 


-0.2243 


+0.4422 


1.10 


+0.2929 


-0.1419 


-0.2193 


+0.3730 


1.20 


+0.2713 


-0.1076 


-0.2129 


+0.3149 


1.30 


+0.2504 


-0.0786 


-0.2054 


+0.2656 


1.40 


+0.2302 


-0.0542 


-0.1971 


+0.2235 


1.50 


+0.2110 


-0.0337 


-0.1882 


+0.1873 


1.60 


+0.1926 


-0.0166 


-0.1788 


+0.1560 


1.70 


+0.1752 


-0.0023 


-0.1692 


+0.1290 


1.80 


+0.1588 


+0.0094 


-0.1594 


+0.1056 


1.90 


+0.1433 


+0.0189 


-0.1496 


+0.0854 


2.00 


+0.1289 


+0.0265 


-0.1399 


+0.0679 


2.10 


+0.1153 


+0.0325 


-0.1304 


+0.0527 


2.20 


+0.1026 


+0.0371 


-0.1210 


+0.0397 


2.30 


+0.0911 


+0.0405 


-0.1120 


+0.0285 


2.40 


+0.0804 


+0.0429 


-0.1032 


+0.0189 


2.50 


+0.0705 


+0.0444 


-0.0948 


+0.0109 


2.60 


+0.0614 


+0.0451 


-0.0868 


+0.0039 


2.70 


+0.0531 


+0.0452 


-0.0791 


-0.0018 


2.80 


+0.0455 


+0.0447 


-0.0719 


-0.0066 


2.90 


+0.0387 


+0.0439 


-0.0650 


-0.0105 


3.00 


+0.0326 


+0.0427 


-0.0586 


-0.0137 


3.10 


+0.0270 


+0.0412 


-0.0526 


-0.0161 


3.20 


+0.0220 


+0.0394 


-0.0469 


-0.0180 


3.30 


+0.0176 


+0.0376 


-0.0417 


-0.0194 


3.40 


+0.0137 


+0.0356 


-0.0369 


-0.0204 


3.50 


+0.0102 


+0.0335 


-0.0325 


-0.0210 


3.60 


+0.0072 


+0.0314 


-0.0284 


-0.0213 


3.70 


+0.0045 


+0.0293 


-0.0246 


-0.0213 


3.80 


+0.0022 


+0.0271 


-0.0212 


-0.0210 


3.90 


+0.0003 


+0.0251 


-0.0180 


-0.0206 



494 



THEORY OF PLATES AND SHELLS 



Table 86. Table of the \^(x) Functions (Continued) 









dtzix) 


d4>i(x) 


X 


^aOc) 


*4<«) 


dx 


dx 


4.00 


-0.0014 


+0.0230 


-0.0152 


-0.0200 


4.10 


-0.0028 


+0.0211 


-0.0127 


-0.0193 


4.20 


-0.0039 


+0.0192 


-0.0104 


-0.0185 


4.30 


-0.0049 


+0.0174 


-0.0083 


-0.0177 


4.40 


-0.0056 


+0.0156 


-0.0065 


-0.0168 


4.50 


-0.0062 


+0.0140 


-0.0049 


-0.0158 


4.60 


-0.0066 


+0.0125 


-0.0035 


-0.0148 


4.70 


-0.0069 


+0.0110 


-0.0023 


-0.0138 


4.80 


-0.0071 


+0.0097 


-0.0012 


-0.0129 


4.90 


-0.0071 


+0.0085 


-0.0003 


-0.0119 


5.00 


-0.0071 


+0.0073 


+0.0005 


-0.0109 


5.10 


-0.0070 


+0.0063 


+0.0012 


-0.0100 


5.20 


-0.0069 


+0.0053 


+0.0017 


-0.0091 


5.30 


-0.0067 


+0.0044 


+0.0022 


-0.0083 


5.40 


-0.0065 


+0.0037 


+0.0025 


-0.0075 


5.50 


-0.0062 


+0.0029 


+0.0028 


-0.0067 


5.60 


-0.0059 


+0.0023 


+0.0030 


-0.0060 


5.70 


-0.0056 


+0.0017 


+0.0032 


-0.0053 


5.80 


-0.0053 


+0.0012 


+0.0033 


-0.0047 


5.90 


-0.0049 


+0.0008 


+0.0033 


-0.0041 


6.00 


-0.0046 


+0.0004 


+0.0033 


-0.0036 



Having solutions (a') and (b f ) of Eq. (r), we conclude that the general solution (n) 
of Eq. (e) is 

w = -~ - -^ [Ci^(2p \Tx) + Crt' 2 (2p \/x) + C 3 *' 3 (2p \G) 



+ C 4 +[(2 P Vx)] (c') 



Numerical values of the functions ^i, . . . , \p A and their first derivatives are given 
in Table 86. l A graphical representation of the functions *//[, . . . , \p A is given in 
Fig. 249. It is seen that the values of these functions increase or decrease rapidly as 
the distance from the end increases. This indicates that in calculating the constants 
of integration in solution (c') we can very often proceed as we did with functions 
(281), i.e., by considering the cylinder as an infinitely long one and using at each 
edge only two of the four constants in solution (c'). 

J This table was calculated by F. Schleicher; see "Kreisplatten auf elastischer 
Unterlage," Berlin, 1926. The well-known Kelvin functions may be used in place 
of the functions \p, to which they relate as follows: ^i(x) = ber x, yp*{x) = — bei x, 
\f/i(x) — — (2/tt) kei x, ^ 4 = — (2/tt) ker x. For more accurate tables of the functions 
under consideration see p. 266. 



GENERAL THEORY OF CYLINDRICAL SHELLS 

ft', W 



495 




In applying the general theory to particular cases, the calculation of the consecutive 
srivatives of w is simplified if we use the following relations: 

*"(*) = -M& --**(& 

here the symbol £ is used in place of 2p V*. From expression (c') we then obtain 

N v = - — u> = - — V»[C,/,(0 + CVitt) + C^',({) + Crt' t (&] (eO 
a a 

r ^ {C,[^,(D - 2^(0] - C,[w-,(€) + 2^(1)] 

* 2x Vi 

+ C[^ 4 (f) - 2*',(*)] - C«[W-,(€) + 2*' 4 (*)]} (/') 



f, = -Z) 



cfc« 



#a 3 



V*{Ci[(0V'.(l) - 4(*)*,(0 +ty\(&\ 



48(1 - v 8 ) 

- C[(€)V'i(€) - 4(*)*i(f) - 8*' t (0] 
+ C,[(*)V«(*) - 4({)^ 4 (€) + S*',i&] 

- Ctiawte) - homo - Mam w) 



dx 



Ea 3 p* 



V^{C,[»,(€). + 2^;({)] 



24(1 - *«) 
+ CMMZ) - 2*' 1 (fl] + C,[«-.(t) + 2*' 4 (fl] + C 4 [W 4 (€) - 2*',(fl]} (A') 



By means of these formulas the deflections and the stresses can be calculated at 
ny point, provided the constants C u . . . , C 4 are determined from the edge condi- 



(296) 



496 THEORY OF PLATES AND SHELLS 

tions. The values of the functions fa, • • • > ^4 and their derivatives are to be taken 
from Table 86 if 2p \/x ^ 6. For larger values of the argument, ths following 
asymptotic expressions are sufficiently accurate: 

'•«>--^r i/v5sta (vH) 
*'•«>- -^ i/v5sta (^ + i) 

As an example, consider a cylindrical tank of the same general dimensions as that 
used in the preceding article (page 487), and assume that the thickness of the wall varies 
horn 14 in. at the bottom to 3£ in. at the top. In such a case the distance of the 
origin of the coordinates (Fig. 248) from the bottom of the tank is 

d + Xo = -§d = 416 in. 

Hence, (2p \Zx) XmmXo+d = 21.45. For such a large value of the argument, the functions 
^1, . . . , ^4 and their first derivatives can be replaced by their asymptotic expressions 
(296). The deflection and the slope at the bottom of the tank corresponding to the 
particular solution (c) are 

- _ IOlL d (dw\ = _ ya* so / ./n 

(wi)*-* 0+ * - Ead + Zo y dx ) x _ xo+d Ea {xo + d) 2 

Considering the length of the cylindrical shell in the axial direction as very large, we 
take the constants C 3 and C 4 in solution (c') as equal to zero and determine the con- 
stants Ci and C 2 so as to make the deflection and the slope at the bottom of the shell 
equal to zero. These requirements give us the two following equations: 



[Cif[(2p V*) + <V 2 (2p V*)].-.+* = ?— 



yd 2 d 



— l —^ {Ci[2p VxM^p Vx) - 2^(2 P V^)] - C 2 [2 P V**i(2p V~x) (?) 

?.x Vx 

, /- 70 2 To 



GENERAL THEORY OF CYLINDRICAL SHELLS 497 

Calculating the values of functions \p h ^2 and their derivatives from the asymptotic 
)rmulas (296) and substituting the resulting values in Eqs. 0*'); we obtain 

C, = -269 -£ N 

&cl y/d + Xo 

C 2 - -299^— yL=N 
-ha Vd + xo 

ihere N = (jr*'^* V2^)t-*i.« 

ubstituting these values of the constants in expression (g r ) we find for the bending 
loment at the bottom 

M = 13,900 lb-in. per in. 

n the same manner, by using expression (Ji r ), we find the magnitude of the shearing 
)rce at the bottom of the tank as 

Qo = 527 lb per in. 

"hese results do not differ much from the values obtained earlier for a tank with 
niform wall thickness (page 487). 

119. Thermal Stresses in Cylindrical Shells. Uniform Temperature 
distribution. If a cylindrical shell with free edges undergoes a uniform 
emperature change, no thermal stresses will be produced. But if the 
dges are supported or clamped, free expansion of the shell is prevented, 
,nd local bending stresses are set up at the edges. Knowing the thermal 
xpansion of a shell when the edges are free, the values of the reactive 
aoments and forces at the edges for any kind of symmetrical support 
an be readily obtained by using Eqs. (279) and (280), as was done in 
he cases shown in Fig. 241. 

Temperature Gradient in the Radial Direction. Assume that h and U 
,re the uniform temperatures of the cylindrical wall at the inside and the 
utside surfaces, respectively, and that the variation of the temperature 
hrough the thickness is linear. In such a case, at points at a large dis- 
ance from the ends of the shell, there will be no bending, and the stresses 
an be calculated by using Eq. (51), which was derived for clamped plates 
see page 50). Thus the stresses at the outer and the inner surfaces are 



<T X = o" 



tp -L- 



± Ea{tl -J£ (a) 



2(1 - v) 



/here the upper sign refers to the outer surface, indicating that a tensile 
tress will act on this surface if t\ > f 2 . 

Near the ends there will usually be some bending of the shell, and the 
otal thermal stresses will be obtained by superposing upon (a) such 
tresses as are necessary to satisfy the boundary conditions. Let us con- 
ider, as an example, the condition of free edges, in which case the stresses 
x must vanish at the ends. In calculating the stresses and deformations 



498 THEORY OF PLATES AND SHELLS 

in this case we observe that at the edge the stresses (a) result in uni- 
formly distributed moments M (Fig. 250a) of the amount 

_ Eajti - hW (b) 

Mo - " 12(1 - v) (6) 

To obtain a free edge, moments of the same magnitude but opposite in 
direction must be superposed (Fig. 2506). Hence the stresses at a free 
edge are obtained by superposing upon the stresses (a) the stresses pro- 
duced by the moments — M (Fig. 2506). These latter stresses can be 
readily calculated by using solution (278). From this solution it follows 
that 

(M x ) x=0 = ^ _ ^ CM,)*-o = v{M x ) x ^ = 12 v (1 _ y) (c) 

It is seen that at the free edge the maximum thermal stress acts in the 
circumferential direction and is obtained by adding to the stress (a) the 
stresses produced by the moment M^ and the force N^. Assuming that 
ii > U, we thus obtain 



^j max - 2(1 - f) V V3 / 



w 



For v = 0.3 this stress is about 25 per cent greater than the stress (a) 
calculated at points at a large distance from the ends. We can therefore 

conclude that if a crack will occur in a brittle 



r^zzzzzzzzzzzzzx x material such as glass due to a temperature 

•rif J difference h — t 2y it will start at the edge and 

2 (a) will proceed in the axial direction. In a similar 

manner the stresses can also be calculated 



M ( (ZZZZZZZZZZZZZZ) — ~ 

m ° X I i n cases in which the edges are clamped or 

,z ( . supported. 1 

„ OCA Temperature Gradient in the Axial Direction. 

Fig. 250 ,.• , . , ,i , ,, 

If the temperature is constant through the 

thickness of the wall but varies along the length of the cylinder, the pro- 
blem can be easily reduced to the solution of Eq. (274). 2 Let t = F(x) be 
the increase of the temperature of the shell from a certain uniform initial 
temperature. Assuming that the shell is divided into infinitely thin rings 
by planes perpendicular to the x axis and denoting the radius of the shell by 
a. the radial expansion of the rings due to the temperature change is aaF(x). 

1 Several examples of this kind are discussed in the paper by C. H. Kent, Trans. 
ASME, vol. 53, p. 167, 1931. 

2 See Timoshenko and Lessells, " Applied Elasticity/' p. 146, 1925. 



GENERAL THEORY OF CYLINDRICAL SHELLS 



499 



This expansion can be eliminated and the shell can be brought to its initial 
diameter by applying an external pressure of an intensity Z such that 



q 2 Z 
Eh 



= aaF(x) 



which gives 



z = E± F(x) 



a 



(/) 



A load of this intensity entirely arrests the thermal expansion of the shell 
and produces in it only circumferential stresses having a magnitude 



Ow = 



aZ 

h 



= -EaF(x) 



(g) 






2a • 



■4- b -*\ 



W 



m 

iitiiliitilA 



To obtain the total thermal stresses, we must superpose on the stresses (g) 
the stresses that will be produced in the shell 
by a load of the intensity — Z. This latter 
load must be applied in order to make the 
lateral surface of the shell free from the ex- 
ternal load given by Eq. (/). The stresses 
produced in the shell by the load — Z are ob- 
tained by the integration of the differential 
equation (276), which in this case becomes 



- 5ZZZZZ2Z2Z2ZZ 



ZZZZZZZZZZZZ2 



d 4 w 
dx* 



+ 4/3<u> = - ^ F(x) (h) 



As an example of the application of this 
equation let us consider a long cylinder, as 
shown in Fig. 251a, and assume that the part 
of the cylinder to the right of the cross section 
mn has a constant temperature to, whereas that 
to the left side has a temperature that decreases linearly to a temperature 
ti at the end x = b according to the relation 




t = t Q - 



(to — ti)x 



The temperature change at a point in this portion is thus 

F{x) =t-t = - {h ~ h)x 



W 



Substituting this expression for the temperature change in Eq. (h), we 
find that the particular solution of that equation is 



Wi 



aa 



(to — h)x 



U) 



500 THEORY OF PLATES AND SHELLS 

The displacement corresponding to this particular solution is shown in 
Fig. 2516, which indicates that there is at the section mn an angle of dis- 
continuity of the magnitude 

? - f e. - « w 

To remove this discontinuity the moments M must be applied. Since 
the stress cr, corresponding to the particular solution (j) cancels the 
stresses (g), we conclude that the stresses produced by the moments M 
are the total thermal stresses resulting from the above-described decrease 
in temperature. If the distances of the cross section ran from the ends 
of the cylinder are large, the magnitude of the moment M can be obtained 
at once from Eq. (280) by substituting 



to obtain 1 



e.=° (&)_--*<••-«■> 



M = ^fiD&{to-ti) (I) 



Substituting for ft its value from expression (275) and taking v = 0.3, 
we find that the maximum thermal stress is 

(0~ = ijr - °- 353 T V ^ (f0 " h) (m) 

It was assumed in this calculation that the length b to the end of the 
cylinder is large. If this is not the case, a correction to the moment (J) 
must be calculated as follows. In an infinitely long shell the moment M 
produces at the distance x = b a moment and a shearing force (Fig. 251c) 2 
that are given by the general solution (282) as 

d*w (n) 

Q x = ~D^= -2/3M f(/36) 

Since at the distance x = b we have a free edge, it is necessary to apply 
there a moment and a force of the magnitude 

— AT, == -Movtfb) -Q x = 2/3ilfof(/3&) (o) 

in order to eliminate the forces (n) (Fig. 251c). 

1 If to — h is positive, as was assumed in the derivation, M is negative and thus 
has the direction shown in Fig. 2516. 

2 The directions M x and Q x shown in Fig. 251c are the positive directions if the 
x axis has the direction shown in Fig. 251a, 



GENERAL THEORY OF CYLINDRICAL SHELLS 501 

The moment produced by the forces (o) at the cross section ran gives 
the desired correction AM which is to be applied to the moment (I). Its 
value can be obtained from the third of the equations (282) if we substi- 
tute in it — M o^(/36) for M and — 2/3ilf f (/36) * for Q . These substitutions 
give 

AM = ~D^ = -M [<p(pb)] 2 - 2Mo[f036)] 2 (p) 

As a numerical example, consider a cast-iron cylinder having the following dimen- 
sions: a = 9|£ in., h = If in., 6 = 4| in.; a = 101 • 10" 7 , £ = 14 • 10 6 psi, 

to - *i = 180°C 
The formula (m) then gives 

ow = 7,720 psi (q) 

In calculating the correction (p), we have 

and, from Table 84, 

vifib) = 0.238 Hfib) = 0.223 
Hence, from Eq. (p), 

AM = -M (0.238 2 + 2 • 0.223 2 ) = -0.156M 

This indicates that the above-calculated maximum stress (q) must be diminished by 
15.6 per cent to obtain the correct maximum value of the thermal stress. 

The method shown here for the calculation of thermal stresses in the case of a linear 
temperature gradient (i) can also be easily applied in cases in which F(x) has other 
than a linear form. 

120. Inextensional Deformation of a Circular Cylindrical Shell. 1 If 

the ends of a thin circular cylindrical shell are free and the loading is not 
symmetrical with respect to the axis of the cylinder, the deformation con- 
sists principally in bending. In such cases the magnitude of deflection 
can be obtained with sufficient accuracy by neglecting entirely the strain 
in the middle surface of the shell. An example of such a loading con- 
dition is shown in Fig. 252. The shortening of the vertical diameter 
along which the forces P act can be found with good accuracy by con- 
sidering only the bending of the shell and assuming that the middle sur- 
face is inextensible. 

Let us first consider the limitations to which the components of dis- 
placement are subject if the deformation of a cylindrical shell is to be 
inextensional. Taking an element in the middle surface of the shell at 
a point O and directing the coordinate axes as shown in Fig. 253, we shall 

* The opposite sign to that in expression (o) is used here, since Eqs. 282 are derived 
for the direction of the x axis opposite to that shown in Fig. 251a. 

1 The theory of inextensional deformations of shells is due to Lord Rayleigh, Proc. 
London Math. Soc., vol. 13, 1881, and Proc. Roy. Soc. (London), vol. 45, 1889. 



502 



THEORY OF PLATES AND SHELLS 



denote by u, v, and w the components in the x, y, and z directions of the 
displacement of the point 0. The strain in the x direction is then 



*z = 



du 
dx 



(a) 



In calculating the strain in the circumferential direction we use Eq. (a) 
(Art. 108, page 446). Thus, 



_ 1 dv __ w 
* a d<p a 

The shearing strain in the middle surface can be expressed by 

__ du dv 

x * a dip dx 



Q>) 



(fi) 



which is the same as in the case of small deflections of plates except that 
a dip takes the place of dy. The condition that the deformation is inexten- 



X 


-+ — 


[ \o 












1 


— 


-- 








.2a 

1 

1 
1 
Y 




pi 


ETh 


, 







Fig. 252 



sional then requires that the three strain components in the middle surface 
must vanish; i.e., 



dx a dtp a 



du , dv _ n 
a dip dx 



(d) 



These requirements are satisfied if we take the displacements in the 
following form : 

ui = 



Vi 



Wi 



= a y (a n cos riip — a' n sin rtip) 

n-l 

00 

= —a y n(a n sin Uip + a' n cos n^>) 



(fi) 



n = l 



where a is the radius of the middle surface of the shell, <p the central 
angle, and a n and a' n constants that must be calculated for each particular 
case of loading. The displacements (e) represent the case in which all 



GENERAL THEORY OF CYLINDRICAL SHELLS 



503 



cross sections of the shell deform identically. On these displacements 
we can superpose displacements two of which vary along the length of 
the cylinder and which are given by the following series: 



U 2 = 



V 2 = X 



W 2 = 



00 

— a y — (& n sin rap + b' n cos n<p) 

71 = 1 

00 

y (b n cos n<p — b' n sin n<p) 

n = l 

00 

x y n(b n sin n<p + b f n cos rap) 



(/) 



It can be readily proved by substitution in Eqs. (d) that these expressions 
also satisfy the conditions of inextensibility. Thus the general expres- 
sions for displacements in inextensional deformation of a cylindrical shell 
are 

u = Ux + u 2 v = Vi + v 2 w = Wi -\- w 2 (g) 

In calculating the inextensional deformations of a cylindrical shell 
under the action of a given system of forces, it is advantageous to use 
the energy method. To establish the required expression for the strain 
energy of bending of the shell, we begin with 
the calculation of the changes of curvature of 
the middle surface of the shell. The change 
of curvature in the direction of the generatrix 
is equal to zero, since, as can be seen from 
expressions (e) and (/), the generatrices re- 
main straight. The change of curvature of 
the circumference is obtained by comparing 
the curvature of an element ran of the circum- 
ference (Fig. 254) before deformation with that of the corresponding ele- 
ment mini after deformation. Before deformation the curvature in the 
circumferential direction is 

dip __ dip _ 1 

ds 







Fig. 254 



a dip 



a 



The curvature of the element mini after deformation is 



dipi d(p+ M ds 



dsi (a — w) dip 



504 THEORY OF PLATES AND SHELLS 

Hence the change in curvature is 

Xv (a - to) d<p a dip a 2 \° + d<p*J 
By using the second of the equations (d) we can also write 

1 /do , d*w\ ,.. 



The bending moment producing this change in curvature is 





m - - R (*L _i_ a ^ > 



and the corresponding strain energy of bending per unit area can be calcu- 
lated as in the discussion of plates (see page 46) and is equal to 

D /dv , d*w\* D ( , dVv 2 ,.,, 

In addition to bending, there will be a twist of each element such as 
that shown at point in Fig. 253. In calculating this twist we note that 
during deformation an element of a generatrix rotates 1 through an angle 
equal to —dw/dx about the y axis and through an angle equal to dv/dx 
about the z axis. Considering a similar element of a generatrix at a 
circumferential distance a dip from the first one, we see that its rotation 
about the y axis, as a result of the displacement w, is 

dw d 2 w , . .. 

~ d^ ~ ^p-Tx d(p b) 

The rotation of the same element in the plane tangent to the shell is 



(£) 



d 
dv v,„, 

di + —dir d<p 

Because of the central angle dip between the two elements, the latter 
rotation has a component with respect to the y axis equal to 2 

~d% d<p (fc) 

From results (j) and (k) we conclude that the total angle of twist between 
the two elements under consideration is 

, / d 2 W , dv\ , 

1 In determining the sign of rotation the right-hand screw rule is used. 
* A small quantity of second order is neglected in this expression. 



GENERAL THEORY OF CYLINDRICAL SHELLS 505 

and that the amount of strain energy per unit area due to twist is (see 
page 47) 

a 2 \d<p dx dxj v 7 

Adding together expressions (i) and (I) and integrating over the surface 
of the shell, the total strain energy of a cylindrical shell undergoing an 
inextensional deformation is found to be 

Substituting for w and v their expressions (g) and integrating, we find for 
a cylinder of a length 21 (Fig. 252) the following expression for strain 
energy : 

oo 

V = * Dl X (U2 a* 1)2 {*' [ a2{al + + I l2{bl + 6 « 2) ] 



+ 2(1 - v)a\bl + ft?) (297) 



71 = 2 



This expression does not contain a term with n = 1, since the corre- 
sponding displacements 

i/'i = a(ai cos <p — a[ sin v?) ( . 

t^i = — a(a x sin <p + a( cos <p) ' 

represent the displacement of the circle in its plane as a rigid body. The 
vertical and horizontal components of this displacement are found by 
substituting <p = tt/2 in expressions (m) to obtain 

(tfi)*«T/2 = —aa[ (wi)^ T/ 2 = —aai 

Such a displacement does not contribute to the strain energy. 

The same conclusion can also be made regarding the displacements 
represented by the terms with n = 1 in expressions (/). 

Let us now apply expression (297) for the strain energy to the calculation of the 
deformation produced in a cylindrical shell by two equal and opposite forces P acting 
along a diameter at a distance c from the middle 1 (Fig. 252). These forces produce 
work only during radial displacements w of their points of application, i.e., at the 
points x = c, <p = 0, and <p = ir. Also, since the terms with coefficients a n and b n 
in the expressions for w\ and w 2 [see Eqs. (e) and (/)] vanish at these points, only terms 
with coefficients a n and b' n will enter in the expression for deformation. By using the 

1 The case of a cylindrical shell reinforced by elastic rings with two opposite forces 
acting along a diameter of every ring was discussed by R. S. Levy, J. Appl. Mechanics, 
vol. 15, p. 30, 1948. 



506 THEORY OF PLATES AND SHELLS 

principle of virtual displacements, the equations for calculating the coefficients a' n 
and b' n are found to be 

— j 8a n = — na 8a n (l + cos nw)P 
da 

dV 

— t db' n = —nc db' n (l + cos nw)P 

Substituting expression (297) for V, we obtain, for the case where n is an even number, 

' - a 2 P 

a * "" " n{n* - \YirDl 

_ „ \ n ) 

, ncPa 3 

bn = "" (n 2 - l)*wDl[inn* + 2(1 - „)a 2 ] 
If n is an odd number, we obtain 

«: = *>: = o <o) 

Hence in this case, from expressions (e) and (/), 

ac cos n^ 



U irDl L( 



71 = 2,4,6,, 



(n 2 - l)%nV + 2(1 - v)a?\ 



Pa* V* f 1 ncx 1 . , . 

#1 = .— > { 1 — - J sin n<p (p) 

wDl Lf (n(n 2 - l) 2 ^ (n 2 - \)%nH* + 2(1 - v)a 2 ]j 

n = 2,4,6, ... 

Pa 3 V I 1 n 2 ^ 

" " *rZ)Z Zy l(n« - l) 2 (n 2 - l) 2 [£n 2 Z 2 + 2(1 - v)a 2 ] 

?i = 2,4,6, . . . 



If the forces P are applied at the middle, c = and the shortening of the vertical 
diameter of the shell is 

2Pa z \^ 1 Pa z 



n = 2,4,0, . 

The increase in the horizontal diameter is 

2Pa z X^ ( — i)n/2+i p a z 

7i = 2,4,6,... 

The change in length of any other diameter can also be readily calculated. The same 
calculations can also be made if c is different from zero, and the deflections vary with 
the distance x from the middle. 

Solution (p) does not satisfy the conditions at the free edges of the shell, since it 
requires the distribution of moments M x = vM+ to prevent any bending in meridional 
planes. This bending is, however, of a local character and does not substantially 
affect the deflections (q) and (r), which are in satisfactory agreement with experiments. 

The method just described for analyzing the inextensional deformation of cylindrical 
shells can also be used in calculating the deformation of a portion of a cylindrical 
shell which is cut from a complete cylinder of radius a by two axial sections making 



GENERAL THEORY OF CYLINDRICAL SHELLS 



507 



an angle a. with one another (Fig. 255). For example, taking for the displacements 
tlie series 



ota \ b n . UlTip 

u — > — sin 

tc Z-/ n a 

a n cos h x / bn cos 

ira \ mr<p Xir \ 

v = / na n sin / no w 

a Z-^ or a Z-/ 



sin 



mnp 




Fig. 255 



we obtain an inextensional deformation of the shell such that the displacements u 

and w and also the bending moments M <p vanish along the edges ran and mini. Such 

conditions are obtained if the shell is 

supported at points m, n, m h ri\ by bars 

directed radially and is loaded by a 

load P in the plane of symmetry. The 

deflection produced by this load can be 

Found by applying the principle of 

virtual displacements. 

121. General Case of Deformation 
3f a Cylindrical Shell. 1 To establish 
the differential equations for the dis- 
placements u, v, and w which define 
the deformation of a shell, we proceed as in the case of plates. We begin 
with the equations of equilibrium of an element cut out from the cylindri- 
cal shell by two adjacent axial sections and by two adjacent sections 
perpendicular to the axis of the cylinder (Fig. 253). The corresponding 
dement of the middle surface of the shell after deformation is shown in 
Fig. 256a and b. In Fig. 256a the resultant forces and in Fig. 2566 the 

1 A general theory of bending of thin shells has been developed by A. E. H. Love; 
lee Phil. Trans. Roy. Soc. (London), ser. A, p. 491, 1888; and his book "Elasticity/' 
tth ed., chap. 24, p. 515, 1927; see also H. Lamb, Proc. London Math. Soc, vol. 21. 
?or bending of cylindrical shells see also H. Reissner, Z. angew. Math. Mech., vol. 13, 
). 133, 1933; L. H. Uonnell, NACA Rept. 479, 1933 (simplified theory); E. Torroja 
ind J. Batanero, "Cubiertos laminares cilindros/' Madrid, 1950; H. Parkus, Osterr. 
ngr.-Arch., vol. 6, p. 30, 1951; W. Zerna, Ingr.-Arch., vol. 20, p. 357, 1952; P. Csonka, 
icta Tech. Acad. Set. Hung., vol. 6, p. 167, 1953. The effect of a concentrated load 
las been considered by A. Aas-Jakobsen, Bauingenieur, vol. 22, p. 343, 1941; by 
{. N. Rabotnov, Doklady Akad. Nauk S.S.S.R., vol. 3, 1946; and by V. Z. Vlasov, 
'A General Theory of Shells/' Moscow, 1949. For cylindrical shells stiffened by 
ibs, see N. J. Hoff, J. Appl. Mechanics, vol. 11, p. 235, 1944; "H. Reissner Anniver- 
ary Volume," Ann Arbor, Mich., 1949; and W. Schnell, Z. Flugwiss., vol. 3, p. 385, 
955. Anisotropic shells (together with a general theory) have been treated by 
V. Fliigge, Ingr.-Arch., vol. 3, p. 463, 1932; also by Vlasov, op. cit., chaps. 11 and 12. 
r or stress distribution around stiffened cutouts, see bibliography in L. S. D. Morley's 
>aper, Natl. Luchtvaarlab. Rappts., p. 362, Amsterdam, 1950. A theory of thick 
ylindrical shells is due to Z. Bazant, Proc. Assoc. Bridge Structural Engrs., vol. 4, 1936. 



508 



THEORY OF PLATES AND SHELLS 



resultant moments, discussed in Art. 104, are shown. Before defor- 
mation, the axes x, y, and z at any point of the middle surface had the 
directions of the generatrix, the tangent to the circumference, and the 
normal to the middle surface of the shell, respectively. After defor- 
mation, which is assumed to be very small, these directions are slightly 
changed. We then take the z axis normal to the deformed middle sur- 
face, the x axis in the direction of a tangent to the generatrix, which may 
have become curved, and the y axis perpendicular to the xz plane. The 




N % + d^ dx ./ 3N 



N 



*9 3x 



*«eb 



N, 






ia) 



P* dy 



dy 




NVJf'dy 



M % + 



^f**^* 



[6) 
Fig. 256 



directions of the resultant forces will also have been slightly changed 
accordingly, and these changes must be considered in writing the equa- 
tions of equilibrium of the element OABC. 

Let us begin by establishing formulas for the angular displacements of 
the sides BC and AB with reference to the sides OA and OC of the ele- 
ment, respectively. In these calculations we consider the displacements 
u, v, and w as very small, calculate the angular motions produced by 
each of these displacements, and obtain the resultant angular displace- 
ment by superposition. We begin with the rotation of the side BC with 
respect to the side OA. This rotation can be resolved into three com- 



GENERAL THEORY OF CYLINDRICAL SHELLS 509 

ponent rotations with respect to the x, y, and z axes. The rotations of 
the sides OA and BC with respect to the x axis are caused by the dis- 
placements v and w. Since the displacements v represent motion of the 
sides OA and BC in the circumferential direction (Fig. 253), if a is the 
radius of the middle surface of the cylinder, the corresponding rotation 
of side OA about the x axis is v/a, and that of side BC is 



l( v + lTx dx ) 



Thus, owing to the displacements v, the relative angular motion of BC 
with respect to OA about the x axis is 

~S-dx (a) 

a dx 

Because of the displacements w, the side OA rotates about the x axis 
through the angle dw/(a d<p), and the side BC through the angle 

dw + * (*l\ dx 



a dcp dx 



\a d<pj 



Thus, because of the displacements w, the relative angular displace- 
ment is 



d / dw\ 

dx \a d(p J 



dx (b) 



Summing up (a) and (6), the relative angular displacement about the 
x axis of side BC with respect to side OA is 

1/dv . d 2 w\, , . 

a \dx dx dip J 

The rotation about the y axis of side BC with respect to side OA is caused 
by bending of the generatrices in axial planes and is equal to 

-w dx (d) 

The rotation about the z axis of side BC with respect to side OA is due to 
bending of the generatrices in tangential planes and is equal to 

B dx (e) 

The formulas (c), (d), and (e) thus give the three components of rotation 
of the side BC with respect to the side OA. 

Let us now establish the corresponding formulas for the angular dis- 
placement of side AB with respect to side OC. Because of the curvature 



510 THEORY OF PLATES AND SHELLS 

of the cylindrical shell, the initial angle between these lateral sides of the 
element OABC is dip. However, because of the displacements v and w 
this angle will be changed. The rotation of the lateral side OC about 
the x axis is 

a^ad<p U) 

The corresponding rotation for the lateral side AB is 

v , dw , d (v , dw \ 7 
a a a^ clip \a a dip J 

Hence, instead of the initial angle dip, we must now use the expression 

In calculating the angle of rotation about the y axis of side AB with 
respect to the side OC we use the expression for twist from the preceding 
article (see page 504) ; this gives the required angular displacement as 



t 



d2w +£)<** w 



K dip dx 

Rotation about the z axis of the side AB with respect to OC is caused by 
the displacements v and w. Because of the displacement v, the angle of 
rotation of side OC is dv/dx, and that of side AB is 

— -4- d / dv\ . 
d.r a d^? \^/ 

so that the relative angular displacement is 

Because of the displacement to, the side AB rotates in the axial plane 
by the angle dw/dx. The component of this rotation with respect to the 
z axis is 

dw , , 

Summing up (i) and (j), the relative angular displacement about the 
z axis of side AB with respect to side OC is 



/ d 2 v _ chA 
\d^> eta dx / 



d^> (fc) 



Having the foregoing formulas 1 for the angles, we may now obtain 
three equations of equilibrium of the element OABC (Fig. 256) by pro- 
jecting all forces on the x, y, and z axes. Beginning with those forces 

1 These formulas can be readily obtained for a cylindrical shell from the general 
formulas given by A. E. H. Love in his book "Elasticity/' 4th ed., p. 523, 1927. 



GENERAL THEORY OF CYLINDRICAL SHELLS 511 

parallel to the resultant forces N z and N^ x and projecting them on the 

x axis, we obtain 

dN x , , dN ¥X , , 

-r— dx a dip — r- 21 dip dx 
dx dip 

Because of the angle of rotation represented by expression (fc), the forces 
parallel to N y give a component in the x direction equal to 

""^ \^~i -z~]d<pdx 



/ d*v _ dw\ 
\d<p dx dx) 



Because of the rotation represented by expression (e), the forces parallel 
bo N ztp give a component in the x direction equal to 



dH 
dx 2 



" N** TZ* dx a dip 



Finally, because of angles represented by expressions (d) and (h), the 
forces parallel to Q x and Q^ give components in the x direction equal to 



-Qx-^dxadip - Q<p(^L + iz) d ^ dx 



dx* ™ " "* ** \d<p dx 



Regarding the external forces acting on the element, we assume that 
there is only a normal pressure of intensity q, the projection of which 
3n the x and y axes is zero. 
Summing up all the projections calculated above, we obtain 

dN *dxad<p + ^d<pdx - N v [^- x - l£)d<pdx 



dx d<p 

dH , , n d*W 

— dxad<p-Q x ^ 



~ N xv ^dxad<p - Q x ^dxad<p - Q*(J^y x + -£) d< P dx = ° 



In the same manner two other equations of equilibrium can be written. 
kiter simplification, all three equations can be put in the following form: 

dN x , dN vx n d 2 w Ar dh) 
>^ + l^- aQ *W- aN **dx-* 

n (Hi a. d * w \ - m ( dH - ?™\ = n 
V v \dx + dxd<p) "'\dxdv dx) 

w, . dN xv . Ar d 2 v n (dv d*w\ 

-r-^ + a -r-* + aN x —„ — Q x ( — + - - ) 
d<p dx dx 2 \dx dx dip J 

, A , / d 2 v dw\ n A dv d*w \ 

+ N -\Wdi -dx-)- Q *\ l + ^r + ^d?) = ° 

l -dx~ + "a7 + M^ + te^) + A a^ 



(298) 



512 THEORY OF PLATES AND SHELLS 

Going now to the three equations of moments with respect to the x, y, 
and z axes (Fig. 2566) and again taking into consideration the small angu- 
lar displacements of the sides BC and AB with respect to OA and OC, 
respectively, we obtain the following equations: 

a •*= - «& - a M, £ - M„ (»L- - £ ) + «<3, = 

dx dip dx 2 \dx dip dx ) 

dM vx dM. M dh>_ /J*_ _ *?S - «Q. - 
_, (dv , d*w\, M d*w , M (. , dv d*w \ 

By using the first two of these equations 1 we can eliminate Q x and Q v 
from Eqs. (298) and obtain in this way three equations containing the 
resultant forces N x , N„ and N xv and the moments M x , M 9t and M xv . 
By using formulas (253) and (254) of Art. 104, all these quantities can be 
expressed in terms of the three strain components e x , e v , and y xv of the 
middle surface and the three curvature changes x*, Xv, and Xx V . By 
using the results of the previous article, these latter quantities can be 
represented in terms of the displacements u, v, and w as follows: 2 



du _ dv _w __ du dv 

* x = Tx ev ~~a~fy a y " v ad?'*' dx 

d*w 1 (dv d*w\ 1 (dv d*w \ 

x ' = to 2 " Xv ~ a 1 \di + ~dVy Xxv « \te dx d <P/ 



(300) 



Thus we finally obtain the three differential equations for the determi- 
nation of the displacements u, v, and w. 

In the derivation equations (298) and (299) the change of curvature 
of the element OABC was taken into consideration. This procedure is 
necessary if the forces N x , N y , and N xv are not small in comparison with 
their critical values, at which lateral buckling of the shell may occur. 3 
If these forces are small, their effect on bending is negligible, and we can 
omit from Eqs. (298) and (299) all terms containing the products of the 
resultant forces or resultant moments with the derivatives of the small 
displacements u, v, and w. In such a case the three Eqs. (298) and the 

1 To satisfy the third of these equations the trapezoidal form of the sides of the 
element OABC must be considered as mentioned in Art. 104. This question is dis- 
cussed by W. Fliigge, "Statik und Dynamik der Schalen," 2d ed., p. 148, Berlin, 1957. 

2 The same expressions for the change of curvature as in the preceding article are 
used, since the effect of strain in the middle surface on curvature is neglected. 

3 The problems of buckling of cylindrical shells are discussed in S. Timoshenko, 
"Theory of Elastic Stability," and will not be considered here. 



GENERAL THEORY OF CYLINDRICAL SHELLS 513 

•st two equations of system (299) can be rewritten in the following 
mplified form: 

dx o<p 

dN* dN x<p n ft 

dcp dx 

dM xv dM v n 

dM vx dM x 

~df + a ~dx-- aQ * = 

lliminating the shearing forces Q x and Q^, we finally obtain the three 
blowing equations: 

dx dtp 

»« s+a »« s+ »« 3 _i»« x . Q (302) 

d(f dx ' dx a dip 

AT d 2 M ipx , d 2 M x d 2 M x<p 1 d 2 M^ 
d.r d<p 3a; 2 3a: 3<p a dip 2 

iy using Eqs. (253), (254), and (300), all the quantities entering in these 
quations can be expressed by the displacements u, v, and w, and we 
btain 

d 2 u 1 — v d 2 u 1 + v d 2 v ^ v dw _ 

dx 2 2a 2 d<p 2 2a dx dip a dx 

+ v d 2 u , I - v d 2 v , \d 2 v 1 dw 
+ a — o — ^—9 + 



2 dx dip 2 dx 2 a dtp 2 a dip 



du dv _w_h 2 / d A w 2 d 4 w d 4 w \ 

' ~dx + aT&p a 12 \ a dx 4 + a dx 2 dip 2 + a 3 dip 4 ) 

_ /^ / 2 - y d 3 *> d 3 z; \ = _ ag(l - v 2 ) 

12 \ a 9a; 2 dip + a 3 d^ 3 / #/i 

More elaborate investigations show 1 that the last two terms on the 
eft-hand side of the second of these equations and the last term on the 
eft-hand side of the third equation are small quantities of the same order 
is those which we already disregarded by assuming a linear distribution 
)f stress through the thickness of the shell and by neglecting the stretch- 
ng of the middle surface of the shell (see page 431). In such a case it 

1 See Vlasov, op. cit. } p. 316, and, for more exact equations, p. 257. 



514 



THEORY OF PLATES AND SHELLS 



will be logical to omit the above-mentioned terms and to use in analysis 
of thin cylindrical shells the following simplified system of equations: 

dh) v dw 



d*u 1 - v d*u 1 + v 

dx 2 2a 2 d<p 2 2a dx dip 



a dx 



= 



1 + v d*u 



du , dv 
v dx a d<p 



dx dtp 

w _ ^1 
a 12 



1 - v dh z 
+ a ° dx* + " *->* 



( a 4 
\ a dx 



2 

w 



iajy 

a d<p 
d*w 



+ l 

dx< a dx' d<p' 



+ 



-1^ = 
a 3 dip 4 ) 



(304) 

aq{\ - v 2 ) 
Eh 



Some simplified expressions for the stress resultants which are in 
accordance with the simplified relations (304) between the displacements 
of the shell will be given in Art. 125. 

From the foregoing it is seen that the problem of a laterally loaded 
cylindrical shell reduces in each particular case to the solution of a sys- 
tem of three differential equations. Several applications of these equa- 
tions will be shown in the next articles. 

122. Cylindrical Shells with Supported Edges. Let us consider the 
case of a cylindrical shell supported at the ends and submitted to the 




Fig. 257 

pressure of an enclosed liquid as shown in Fig. 257. l The conditions at 
the supports and the conditions of symmetry of deformation will be 
satisfied if we take the components of displacement in the form of the 
following series: 

A lUTTX 

U = } ) Amn COS Uip COS 



-n 



B m n sin ri(p sin 



C mn cos rap sin 



I 

7YVKX 

mwx 



(a) 



in which I is the length of the cylinder and <p is the angle measured as 
shown in Fig. 257. 2 

*See S. Timoshenko, " Theory of Elasticity," vol. 2, p. 385, St. Petersburg, 1916 
(Russian). 

2 By substituting expressions (a) in Eqs. (300) it can be shown that the tensile 
forces N x and the moments M x vanish at the ends; the shearing forces do not vanish, 
however, since y x<p and M Xip are not zero at the ends. 



GENERAL THEORY OF CYLINDRICAL SHELLS 515 

The intensity of the load q is represented by the following expressions: 

q = — 7a(cos <p — cos a) when <p < a 

q = when <p > a ' ' 

1 which 7 is the specific weight of the liquid and the angle a defines the 
>vel of the liquid, as shown in Fig. 2576. The load q can be represented 
y the series 



-2I» 



mn cos n<p sin —y— (c) 



i which the coefficients D mn can be readily calculated in the usual way 
rom expressions (6). These coefficients are represented by the expression 

L>mn = 2 / 2 — 1 n (cos a sin na — 7i cos na sm a) (a) 

/here m = 1, 3, 5, . . . and n = 2, 3, 4, . . . 

whereas D m0 = « (sin a — a cos a) (e) 

*d D ml = - ^ (2a - sin 2a) (/) 

m7r 2 / VJ/ 

n the case of a cylindrical shell completely filled with liquid, we denote 
he pressure at the axis of the cylinder 1 by yd; then 

q = —y(d + a cos v) (g) 

tnd we obtain, instead of expressions (d), (e), and (/), 

D mn - D m0 = - ^ D _ Aya 

mir mir v ' 

To obtain the deformation of the shell we substitute expressions (a) 
ind (c) in Eqs. (304). In this way we obtain for each pair of values of 
n and n a system of three linear equations from which the corresponding 
values of the coefficients A mn , B mn , and C mn can be calculated. 2 Taking a 
particular case in which d = a, we find that for n = and m = 1, 3, 5, . . . 
these equations are especially simple, and we obtain 

p — n n _ mir a ^N 

AV 



where N = ^^ X = L ,= 



2yal 2 h __ I 

D A ~ a ''~2l 



3m Fx*(l - v 2 ) + ^ m<r 4 l 



1 In a closed cylindrical vessel this pressure can be larger than ay. 

* Such calculations have been made for several particular cases by I. A. Wojtaszak, 
Phil. Mag., ser. 7, vol. 18, p. 1099, 1934; see also the paper by H. Reissner in Z. angew. 
Math. Mech., vol. 13, p. 133, 1933, 



516 



THEORY OF PLATES AND SHELLS 



For n = 1 the expressions for the coefficients are more complicated. To 
show how rapidly the coefficients diminish as m increases, we include in 
Table 87 the numerical values of the coefficients for a particular case in 
which a = 50 cm, I = 25 cm, h = 7 cm, v = 0.3, and a = tt. 

Table 87. The Values of the Coefficients in Expressions (a) 



m 


2-103 
Nh 


2-10' 

Cm0 Nh 


2-103 


2- 103 


2- 10' 

Cwl Nh 


1 
3 
5 


57.88 
0.1073 
0.00503 


-1,212. 
-6.742 
-0.526 


49.18 
0.1051 
0.00499 


-66.26 
-0.0432 
-0.00122 


-1,183 

6.704 
0.525 



It is seen that the coefficients rapidly diminish as m increases. Hence, 

by limiting the number of coeffi- 
cients to those given in the table, 
we shall obtain the deformation of 
the shell with satisfactory accuracy. 
123. Deflection of a Portion of 
a Cylindrical Shell. The method 
used in the preceding article can 
also be applied to a portion of a 
cylindrical shell which is supported 
along the edges and submitted to 
the action of a uniformly distrib- 
uted load q normal to the surface 
(Fig. 258). l We take the compo- 
nents of displacement in the form of the series 




Fig. 258 



u 






A . U-Kip 7YVKX 

A mn sin cos — t— 

a I 

rmrx 



B mn cos sin 



a 



I 



(a) 



y . ninp . rmrx 
, mn sin sin 



a 



I 



in which a is the central angle subtended by the shell and I is the length 
of the shell. It can be shown by substitution of expressions (a) in Eqs. 
(300) that in this way we shall satisfy the conditions at the boundary, 
which require that along the edges <p = and cp = a the deflection w, 
the force N 99 and the moment M v vanish and that along the edges x = 
and x = I the deflection w } the force N x , and the moment M x vanish. 

1 See Timoshenko, "Theory of Elasticity/' vol. 2, p. 386, 1916. 



GENERAL THEORY OF CYLINDRICAL SHELLS 517 

ie intensity of the normal load q can be represented by the series 



■ii' 



. mr<p . rrnrx /, N 

mn sin sin — y— (6) 



a 



I 



ibstituting series (a) and (6) in Eqs. (304), we obtain the following sys- 
m of linear algebraic equations for calculating the coefficients A mny B mn , 
id C mw " 

r/am\ 2 (1 - v)n* ~\ (1 + v)amn r vam _ ft 

ii-» ^ T j + 2a2 j + 5 mn x — ^ — + c --r - ° 



. (1 + v)amn , „ [ (1 — v)a 2 m 2 n^~| 



n 



+ C m » - = (c) 



a 



i^m^ + ^Y D - 



am ' nj . ^ f^ 7r 4 /i 2 /a 2 m 2 ^ 2 Vl_n o 2 (l — " 2 ) 

»w -5 — r /5 mn hU 



I T mB a ' " mn [ ' 12a 2 \ Z 2 ' a 2 / J mn #/i 

To illustrate the application of these equations let us consider the case 
a uniformly distributed load 1 acting on a portion of a cylindrical shell 

iving a small angle a and a small sag / = a[l — cos (a/2)]. In this 

irticular case expression (6) becomes 

16a . rmrx . rnra /JX 

sin — =— sin yd) 



= y y igg 

Zy A/ 7r 2 ?nn 



r 2 ?nn Z a 

1,3,5,... 1,3,5," 

id the coefficients D mn are given by the expression 

D mn =™L (e) 

ra?Z7r 2 

abstituting these values in Eqs. (c), we can calculate the coefficients 
mn, B mny and C mn . The calculations made fo